tor-browser

btree.h (119548B)

---

// Copyright 2018 The Abseil Authors.
//
// Licensed under the Apache License, Version 2.0 (the "License");
// you may not use this file except in compliance with the License.
// You may obtain a copy of the License at
//
//      https://www.apache.org/licenses/LICENSE-2.0
//
// Unless required by applicable law or agreed to in writing, software
// distributed under the License is distributed on an "AS IS" BASIS,
// WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
// See the License for the specific language governing permissions and
// limitations under the License.

// A btree implementation of the STL set and map interfaces. A btree is smaller
// and generally also faster than STL set/map (refer to the benchmarks below).
// The red-black tree implementation of STL set/map has an overhead of 3
// pointers (left, right and parent) plus the node color information for each
// stored value. So a set<int32_t> consumes 40 bytes for each value stored in
// 64-bit mode. This btree implementation stores multiple values on fixed
// size nodes (usually 256 bytes) and doesn't store child pointers for leaf
// nodes. The result is that a btree_set<int32_t> may use much less memory per
// stored value. For the random insertion benchmark in btree_bench.cc, a
// btree_set<int32_t> with node-size of 256 uses 5.1 bytes per stored value.
//
// The packing of multiple values on to each node of a btree has another effect
// besides better space utilization: better cache locality due to fewer cache
// lines being accessed. Better cache locality translates into faster
// operations.
//
// CAVEATS
//
// Insertions and deletions on a btree can cause splitting, merging or
// rebalancing of btree nodes. And even without these operations, insertions
// and deletions on a btree will move values around within a node. In both
// cases, the result is that insertions and deletions can invalidate iterators
// pointing to values other than the one being inserted/deleted. Therefore, this
// container does not provide pointer stability. This is notably different from
// STL set/map which takes care to not invalidate iterators on insert/erase
// except, of course, for iterators pointing to the value being erased.  A
// partial workaround when erasing is available: erase() returns an iterator
// pointing to the item just after the one that was erased (or end() if none
// exists).

#ifndef ABSL_CONTAINER_INTERNAL_BTREE_H_
#define ABSL_CONTAINER_INTERNAL_BTREE_H_

#include <algorithm>
#include <cassert>
#include <cstddef>
#include <cstdint>
#include <cstring>
#include <functional>
#include <iterator>
#include <limits>
#include <string>
#include <type_traits>
#include <utility>

#include "absl/base/config.h"
#include "absl/base/internal/raw_logging.h"
#include "absl/base/macros.h"
#include "absl/base/optimization.h"
#include "absl/container/internal/common.h"
#include "absl/container/internal/common_policy_traits.h"
#include "absl/container/internal/compressed_tuple.h"
#include "absl/container/internal/container_memory.h"
#include "absl/container/internal/layout.h"
#include "absl/memory/memory.h"
#include "absl/meta/type_traits.h"
#include "absl/strings/cord.h"
#include "absl/strings/string_view.h"
#include "absl/types/compare.h"

namespace absl {
ABSL_NAMESPACE_BEGIN
namespace container_internal {

#ifdef ABSL_BTREE_ENABLE_GENERATIONS
#error ABSL_BTREE_ENABLE_GENERATIONS cannot be directly set
#elif (defined(ABSL_HAVE_ADDRESS_SANITIZER) ||   \
      defined(ABSL_HAVE_HWADDRESS_SANITIZER) || \
      defined(ABSL_HAVE_MEMORY_SANITIZER)) &&   \
   !defined(NDEBUG_SANITIZER)  // If defined, performance is important.
// When compiled in sanitizer mode, we add generation integers to the nodes and
// iterators. When iterators are used, we validate that the container has not
// been mutated since the iterator was constructed.
#define ABSL_BTREE_ENABLE_GENERATIONS
#endif

#ifdef ABSL_BTREE_ENABLE_GENERATIONS
constexpr bool BtreeGenerationsEnabled() { return true; }
#else
constexpr bool BtreeGenerationsEnabled() { return false; }
#endif

template <typename Compare, typename T, typename U>
using compare_result_t = absl::result_of_t<const Compare(const T &, const U &)>;

// A helper class that indicates if the Compare parameter is a key-compare-to
// comparator.
template <typename Compare, typename T>
using btree_is_key_compare_to =
   std::is_convertible<compare_result_t<Compare, T, T>, absl::weak_ordering>;

struct StringBtreeDefaultLess {
 using is_transparent = void;

 StringBtreeDefaultLess() = default;

 // Compatibility constructor.
 StringBtreeDefaultLess(std::less<std::string>) {}        // NOLINT
 StringBtreeDefaultLess(std::less<absl::string_view>) {}  // NOLINT

 // Allow converting to std::less for use in key_comp()/value_comp().
 explicit operator std::less<std::string>() const { return {}; }
 explicit operator std::less<absl::string_view>() const { return {}; }
 explicit operator std::less<absl::Cord>() const { return {}; }

 absl::weak_ordering operator()(absl::string_view lhs,
                                absl::string_view rhs) const {
   return compare_internal::compare_result_as_ordering(lhs.compare(rhs));
 }
 StringBtreeDefaultLess(std::less<absl::Cord>) {}  // NOLINT
 absl::weak_ordering operator()(const absl::Cord &lhs,
                                const absl::Cord &rhs) const {
   return compare_internal::compare_result_as_ordering(lhs.Compare(rhs));
 }
 absl::weak_ordering operator()(const absl::Cord &lhs,
                                absl::string_view rhs) const {
   return compare_internal::compare_result_as_ordering(lhs.Compare(rhs));
 }
 absl::weak_ordering operator()(absl::string_view lhs,
                                const absl::Cord &rhs) const {
   return compare_internal::compare_result_as_ordering(-rhs.Compare(lhs));
 }
};

struct StringBtreeDefaultGreater {
 using is_transparent = void;

 StringBtreeDefaultGreater() = default;

 StringBtreeDefaultGreater(std::greater<std::string>) {}        // NOLINT
 StringBtreeDefaultGreater(std::greater<absl::string_view>) {}  // NOLINT

 // Allow converting to std::greater for use in key_comp()/value_comp().
 explicit operator std::greater<std::string>() const { return {}; }
 explicit operator std::greater<absl::string_view>() const { return {}; }
 explicit operator std::greater<absl::Cord>() const { return {}; }

 absl::weak_ordering operator()(absl::string_view lhs,
                                absl::string_view rhs) const {
   return compare_internal::compare_result_as_ordering(rhs.compare(lhs));
 }
 StringBtreeDefaultGreater(std::greater<absl::Cord>) {}  // NOLINT
 absl::weak_ordering operator()(const absl::Cord &lhs,
                                const absl::Cord &rhs) const {
   return compare_internal::compare_result_as_ordering(rhs.Compare(lhs));
 }
 absl::weak_ordering operator()(const absl::Cord &lhs,
                                absl::string_view rhs) const {
   return compare_internal::compare_result_as_ordering(-lhs.Compare(rhs));
 }
 absl::weak_ordering operator()(absl::string_view lhs,
                                const absl::Cord &rhs) const {
   return compare_internal::compare_result_as_ordering(rhs.Compare(lhs));
 }
};

// See below comments for checked_compare.
template <typename Compare, bool is_class = std::is_class<Compare>::value>
struct checked_compare_base : Compare {
 using Compare::Compare;
 explicit checked_compare_base(Compare c) : Compare(std::move(c)) {}
 const Compare &comp() const { return *this; }
};
template <typename Compare>
struct checked_compare_base<Compare, false> {
 explicit checked_compare_base(Compare c) : compare(std::move(c)) {}
 const Compare &comp() const { return compare; }
 Compare compare;
};

// A mechanism for opting out of checked_compare for use only in btree_test.cc.
struct BtreeTestOnlyCheckedCompareOptOutBase {};

// A helper class to adapt the specified comparator for two use cases:
// (1) When using common Abseil string types with common comparison functors,
// convert a boolean comparison into a three-way comparison that returns an
// `absl::weak_ordering`. This helper class is specialized for
// less<std::string>, greater<std::string>, less<string_view>,
// greater<string_view>, less<absl::Cord>, and greater<absl::Cord>.
// (2) Adapt the comparator to diagnose cases of non-strict-weak-ordering (see
// https://en.cppreference.com/w/cpp/named_req/Compare) in debug mode. Whenever
// a comparison is made, we will make assertions to verify that the comparator
// is valid.
template <typename Compare, typename Key>
struct key_compare_adapter {
 // Inherit from checked_compare_base to support function pointers and also
 // keep empty-base-optimization (EBO) support for classes.
 // Note: we can't use CompressedTuple here because that would interfere
 // with the EBO for `btree::rightmost_`. `btree::rightmost_` is itself a
 // CompressedTuple and nested `CompressedTuple`s don't support EBO.
 // TODO(b/214288561): use CompressedTuple instead once it supports EBO for
 // nested `CompressedTuple`s.
 struct checked_compare : checked_compare_base<Compare> {
  private:
   using Base = typename checked_compare::checked_compare_base;
   using Base::comp;

   // If possible, returns whether `t` is equivalent to itself. We can only do
   // this for `Key`s because we can't be sure that it's safe to call
   // `comp()(k, k)` otherwise. Even if SFINAE allows it, there could be a
   // compilation failure inside the implementation of the comparison operator.
   bool is_self_equivalent(const Key &k) const {
     // Note: this works for both boolean and three-way comparators.
     return comp()(k, k) == 0;
   }
   // If we can't compare `t` with itself, returns true unconditionally.
   template <typename T>
   bool is_self_equivalent(const T &) const {
     return true;
   }

  public:
   using Base::Base;
   checked_compare(Compare comp) : Base(std::move(comp)) {}  // NOLINT

   // Allow converting to Compare for use in key_comp()/value_comp().
   explicit operator Compare() const { return comp(); }

   template <typename T, typename U,
             absl::enable_if_t<
                 std::is_same<bool, compare_result_t<Compare, T, U>>::value,
                 int> = 0>
   bool operator()(const T &lhs, const U &rhs) const {
     // NOTE: if any of these assertions fail, then the comparator does not
     // establish a strict-weak-ordering (see
     // https://en.cppreference.com/w/cpp/named_req/Compare).
     assert(is_self_equivalent(lhs));
     assert(is_self_equivalent(rhs));
     const bool lhs_comp_rhs = comp()(lhs, rhs);
     assert(!lhs_comp_rhs || !comp()(rhs, lhs));
     return lhs_comp_rhs;
   }

   template <
       typename T, typename U,
       absl::enable_if_t<std::is_convertible<compare_result_t<Compare, T, U>,
                                             absl::weak_ordering>::value,
                         int> = 0>
   absl::weak_ordering operator()(const T &lhs, const U &rhs) const {
     // NOTE: if any of these assertions fail, then the comparator does not
     // establish a strict-weak-ordering (see
     // https://en.cppreference.com/w/cpp/named_req/Compare).
     assert(is_self_equivalent(lhs));
     assert(is_self_equivalent(rhs));
     const absl::weak_ordering lhs_comp_rhs = comp()(lhs, rhs);
#ifndef NDEBUG
     const absl::weak_ordering rhs_comp_lhs = comp()(rhs, lhs);
     if (lhs_comp_rhs > 0) {
       assert(rhs_comp_lhs < 0 && "lhs_comp_rhs > 0 -> rhs_comp_lhs < 0");
     } else if (lhs_comp_rhs == 0) {
       assert(rhs_comp_lhs == 0 && "lhs_comp_rhs == 0 -> rhs_comp_lhs == 0");
     } else {
       assert(rhs_comp_lhs > 0 && "lhs_comp_rhs < 0 -> rhs_comp_lhs > 0");
     }
#endif
     return lhs_comp_rhs;
   }
 };
 using type = absl::conditional_t<
     std::is_base_of<BtreeTestOnlyCheckedCompareOptOutBase, Compare>::value,
     Compare, checked_compare>;
};

template <>
struct key_compare_adapter<std::less<std::string>, std::string> {
 using type = StringBtreeDefaultLess;
};

template <>
struct key_compare_adapter<std::greater<std::string>, std::string> {
 using type = StringBtreeDefaultGreater;
};

template <>
struct key_compare_adapter<std::less<absl::string_view>, absl::string_view> {
 using type = StringBtreeDefaultLess;
};

template <>
struct key_compare_adapter<std::greater<absl::string_view>, absl::string_view> {
 using type = StringBtreeDefaultGreater;
};

template <>
struct key_compare_adapter<std::less<absl::Cord>, absl::Cord> {
 using type = StringBtreeDefaultLess;
};

template <>
struct key_compare_adapter<std::greater<absl::Cord>, absl::Cord> {
 using type = StringBtreeDefaultGreater;
};

// Detects an 'absl_btree_prefer_linear_node_search' member. This is
// a protocol used as an opt-in or opt-out of linear search.
//
//  For example, this would be useful for key types that wrap an integer
//  and define their own cheap operator<(). For example:
//
//   class K {
//    public:
//     using absl_btree_prefer_linear_node_search = std::true_type;
//     ...
//    private:
//     friend bool operator<(K a, K b) { return a.k_ < b.k_; }
//     int k_;
//   };
//
//   btree_map<K, V> m;  // Uses linear search
//
// If T has the preference tag, then it has a preference.
// Btree will use the tag's truth value.
template <typename T, typename = void>
struct has_linear_node_search_preference : std::false_type {};
template <typename T, typename = void>
struct prefers_linear_node_search : std::false_type {};
template <typename T>
struct has_linear_node_search_preference<
   T, absl::void_t<typename T::absl_btree_prefer_linear_node_search>>
   : std::true_type {};
template <typename T>
struct prefers_linear_node_search<
   T, absl::void_t<typename T::absl_btree_prefer_linear_node_search>>
   : T::absl_btree_prefer_linear_node_search {};

template <typename Compare, typename Key>
constexpr bool compare_has_valid_result_type() {
 using compare_result_type = compare_result_t<Compare, Key, Key>;
 return std::is_same<compare_result_type, bool>::value ||
        std::is_convertible<compare_result_type, absl::weak_ordering>::value;
}

template <typename original_key_compare, typename value_type>
class map_value_compare {
 template <typename Params>
 friend class btree;

 // Note: this `protected` is part of the API of std::map::value_compare. See
 // https://en.cppreference.com/w/cpp/container/map/value_compare.
protected:
 explicit map_value_compare(original_key_compare c) : comp(std::move(c)) {}

 original_key_compare comp;  // NOLINT

public:
 auto operator()(const value_type &lhs, const value_type &rhs) const
     -> decltype(comp(lhs.first, rhs.first)) {
   return comp(lhs.first, rhs.first);
 }
};

template <typename Key, typename Compare, typename Alloc, int TargetNodeSize,
         bool IsMulti, bool IsMap, typename SlotPolicy>
struct common_params : common_policy_traits<SlotPolicy> {
 using original_key_compare = Compare;

 // If Compare is a common comparator for a string-like type, then we adapt it
 // to use heterogeneous lookup and to be a key-compare-to comparator.
 // We also adapt the comparator to diagnose invalid comparators in debug mode.
 // We disable this when `Compare` is invalid in a way that will cause
 // adaptation to fail (having invalid return type) so that we can give a
 // better compilation failure in static_assert_validation. If we don't do
 // this, then there will be cascading compilation failures that are confusing
 // for users.
 using key_compare =
     absl::conditional_t<!compare_has_valid_result_type<Compare, Key>(),
                         Compare,
                         typename key_compare_adapter<Compare, Key>::type>;

 static constexpr bool kIsKeyCompareStringAdapted =
     std::is_same<key_compare, StringBtreeDefaultLess>::value ||
     std::is_same<key_compare, StringBtreeDefaultGreater>::value;
 static constexpr bool kIsKeyCompareTransparent =
     IsTransparent<original_key_compare>::value || kIsKeyCompareStringAdapted;

 // A type which indicates if we have a key-compare-to functor or a plain old
 // key-compare functor.
 using is_key_compare_to = btree_is_key_compare_to<key_compare, Key>;

 using allocator_type = Alloc;
 using key_type = Key;
 using size_type = size_t;
 using difference_type = ptrdiff_t;

 using slot_policy = SlotPolicy;
 using slot_type = typename slot_policy::slot_type;
 using value_type = typename slot_policy::value_type;
 using init_type = typename slot_policy::mutable_value_type;
 using pointer = value_type *;
 using const_pointer = const value_type *;
 using reference = value_type &;
 using const_reference = const value_type &;

 using value_compare =
     absl::conditional_t<IsMap,
                         map_value_compare<original_key_compare, value_type>,
                         original_key_compare>;
 using is_map_container = std::integral_constant<bool, IsMap>;

 // For the given lookup key type, returns whether we can have multiple
 // equivalent keys in the btree. If this is a multi-container, then we can.
 // Otherwise, we can have multiple equivalent keys only if all of the
 // following conditions are met:
 // - The comparator is transparent.
 // - The lookup key type is not the same as key_type.
 // - The comparator is not a StringBtreeDefault{Less,Greater} comparator
 //   that we know has the same equivalence classes for all lookup types.
 template <typename LookupKey>
 constexpr static bool can_have_multiple_equivalent_keys() {
   return IsMulti || (IsTransparent<key_compare>::value &&
                      !std::is_same<LookupKey, Key>::value &&
                      !kIsKeyCompareStringAdapted);
 }

 enum {
   kTargetNodeSize = TargetNodeSize,

   // Upper bound for the available space for slots. This is largest for leaf
   // nodes, which have overhead of at least a pointer + 4 bytes (for storing
   // 3 field_types and an enum).
   kNodeSlotSpace = TargetNodeSize - /*minimum overhead=*/(sizeof(void *) + 4),
 };

 // This is an integral type large enough to hold as many slots as will fit a
 // node of TargetNodeSize bytes.
 using node_count_type =
     absl::conditional_t<(kNodeSlotSpace / sizeof(slot_type) >
                          (std::numeric_limits<uint8_t>::max)()),
                         uint16_t, uint8_t>;  // NOLINT
};

// An adapter class that converts a lower-bound compare into an upper-bound
// compare. Note: there is no need to make a version of this adapter specialized
// for key-compare-to functors because the upper-bound (the first value greater
// than the input) is never an exact match.
template <typename Compare>
struct upper_bound_adapter {
 explicit upper_bound_adapter(const Compare &c) : comp(c) {}
 template <typename K1, typename K2>
 bool operator()(const K1 &a, const K2 &b) const {
   // Returns true when a is not greater than b.
   return !compare_internal::compare_result_as_less_than(comp(b, a));
 }

private:
 Compare comp;
};

enum class MatchKind : uint8_t { kEq, kNe };

template <typename V, bool IsCompareTo>
struct SearchResult {
 V value;
 MatchKind match;

 static constexpr bool HasMatch() { return true; }
 bool IsEq() const { return match == MatchKind::kEq; }
};

// When we don't use CompareTo, `match` is not present.
// This ensures that callers can't use it accidentally when it provides no
// useful information.
template <typename V>
struct SearchResult<V, false> {
 SearchResult() = default;
 explicit SearchResult(V v) : value(v) {}
 SearchResult(V v, MatchKind /*match*/) : value(v) {}

 V value;

 static constexpr bool HasMatch() { return false; }
 static constexpr bool IsEq() { return false; }
};

// A node in the btree holding. The same node type is used for both internal
// and leaf nodes in the btree, though the nodes are allocated in such a way
// that the children array is only valid in internal nodes.
template <typename Params>
class btree_node {
 using is_key_compare_to = typename Params::is_key_compare_to;
 using field_type = typename Params::node_count_type;
 using allocator_type = typename Params::allocator_type;
 using slot_type = typename Params::slot_type;
 using original_key_compare = typename Params::original_key_compare;

public:
 using params_type = Params;
 using key_type = typename Params::key_type;
 using value_type = typename Params::value_type;
 using pointer = typename Params::pointer;
 using const_pointer = typename Params::const_pointer;
 using reference = typename Params::reference;
 using const_reference = typename Params::const_reference;
 using key_compare = typename Params::key_compare;
 using size_type = typename Params::size_type;
 using difference_type = typename Params::difference_type;

 // Btree decides whether to use linear node search as follows:
 //   - If the comparator expresses a preference, use that.
 //   - If the key expresses a preference, use that.
 //   - If the key is arithmetic and the comparator is std::less or
 //     std::greater, choose linear.
 //   - Otherwise, choose binary.
 // TODO(ezb): Might make sense to add condition(s) based on node-size.
 using use_linear_search = std::integral_constant<
     bool, has_linear_node_search_preference<original_key_compare>::value
               ? prefers_linear_node_search<original_key_compare>::value
           : has_linear_node_search_preference<key_type>::value
               ? prefers_linear_node_search<key_type>::value
               : std::is_arithmetic<key_type>::value &&
                     (std::is_same<std::less<key_type>,
                                   original_key_compare>::value ||
                      std::is_same<std::greater<key_type>,
                                   original_key_compare>::value)>;

 // This class is organized by absl::container_internal::Layout as if it had
 // the following structure:
 //   // A pointer to the node's parent.
 //   btree_node *parent;
 //
 //   // When ABSL_BTREE_ENABLE_GENERATIONS is defined, we also have a
 //   // generation integer in order to check that when iterators are
 //   // used, they haven't been invalidated already. Only the generation on
 //   // the root is used, but we have one on each node because whether a node
 //   // is root or not can change.
 //   uint32_t generation;
 //
 //   // The position of the node in the node's parent.
 //   field_type position;
 //   // The index of the first populated value in `values`.
 //   // TODO(ezb): right now, `start` is always 0. Update insertion/merge
 //   // logic to allow for floating storage within nodes.
 //   field_type start;
 //   // The index after the last populated value in `values`. Currently, this
 //   // is the same as the count of values.
 //   field_type finish;
 //   // The maximum number of values the node can hold. This is an integer in
 //   // [1, kNodeSlots] for root leaf nodes, kNodeSlots for non-root leaf
 //   // nodes, and kInternalNodeMaxCount (as a sentinel value) for internal
 //   // nodes (even though there are still kNodeSlots values in the node).
 //   // TODO(ezb): make max_count use only 4 bits and record log2(capacity)
 //   // to free extra bits for is_root, etc.
 //   field_type max_count;
 //
 //   // The array of values. The capacity is `max_count` for leaf nodes and
 //   // kNodeSlots for internal nodes. Only the values in
 //   // [start, finish) have been initialized and are valid.
 //   slot_type values[max_count];
 //
 //   // The array of child pointers. The keys in children[i] are all less
 //   // than key(i). The keys in children[i + 1] are all greater than key(i).
 //   // There are 0 children for leaf nodes and kNodeSlots + 1 children for
 //   // internal nodes.
 //   btree_node *children[kNodeSlots + 1];
 //
 // This class is only constructed by EmptyNodeType. Normally, pointers to the
 // layout above are allocated, cast to btree_node*, and de-allocated within
 // the btree implementation.
 ~btree_node() = default;
 btree_node(btree_node const &) = delete;
 btree_node &operator=(btree_node const &) = delete;

protected:
 btree_node() = default;

private:
 using layout_type =
     absl::container_internal::Layout<btree_node *, uint32_t, field_type,
                                      slot_type, btree_node *>;
 using leaf_layout_type = typename layout_type::template WithStaticSizes<
     /*parent*/ 1,
     /*generation*/ BtreeGenerationsEnabled() ? 1 : 0,
     /*position, start, finish, max_count*/ 4>;
 constexpr static size_type SizeWithNSlots(size_type n) {
   return leaf_layout_type(/*slots*/ n, /*children*/ 0).AllocSize();
 }
 // A lower bound for the overhead of fields other than slots in a leaf node.
 constexpr static size_type MinimumOverhead() {
   return SizeWithNSlots(1) - sizeof(slot_type);
 }

 // Compute how many values we can fit onto a leaf node taking into account
 // padding.
 constexpr static size_type NodeTargetSlots(const size_type begin,
                                            const size_type end) {
   return begin == end ? begin
          : SizeWithNSlots((begin + end) / 2 + 1) >
                  params_type::kTargetNodeSize
              ? NodeTargetSlots(begin, (begin + end) / 2)
              : NodeTargetSlots((begin + end) / 2 + 1, end);
 }

 constexpr static size_type kTargetNodeSize = params_type::kTargetNodeSize;
 constexpr static size_type kNodeTargetSlots =
     NodeTargetSlots(0, kTargetNodeSize);

 // We need a minimum of 3 slots per internal node in order to perform
 // splitting (1 value for the two nodes involved in the split and 1 value
 // propagated to the parent as the delimiter for the split). For performance
 // reasons, we don't allow 3 slots-per-node due to bad worst case occupancy of
 // 1/3 (for a node, not a b-tree).
 constexpr static size_type kMinNodeSlots = 4;

 constexpr static size_type kNodeSlots =
     kNodeTargetSlots >= kMinNodeSlots ? kNodeTargetSlots : kMinNodeSlots;

 using internal_layout_type = typename layout_type::template WithStaticSizes<
     /*parent*/ 1,
     /*generation*/ BtreeGenerationsEnabled() ? 1 : 0,
     /*position, start, finish, max_count*/ 4, /*slots*/ kNodeSlots,
     /*children*/ kNodeSlots + 1>;

 // The node is internal (i.e. is not a leaf node) if and only if `max_count`
 // has this value.
 constexpr static field_type kInternalNodeMaxCount = 0;

 // Leaves can have less than kNodeSlots values.
 constexpr static leaf_layout_type LeafLayout(
     const size_type slot_count = kNodeSlots) {
   return leaf_layout_type(slot_count, 0);
 }
 constexpr static auto InternalLayout() { return internal_layout_type(); }
 constexpr static size_type LeafSize(const size_type slot_count = kNodeSlots) {
   return LeafLayout(slot_count).AllocSize();
 }
 constexpr static size_type InternalSize() {
   return InternalLayout().AllocSize();
 }

 constexpr static size_type Alignment() {
   static_assert(LeafLayout(1).Alignment() == InternalLayout().Alignment(),
                 "Alignment of all nodes must be equal.");
   return InternalLayout().Alignment();
 }

 // N is the index of the type in the Layout definition.
 // ElementType<N> is the Nth type in the Layout definition.
 template <size_type N>
 inline typename layout_type::template ElementType<N> *GetField() {
   // We assert that we don't read from values that aren't there.
   assert(N < 4 || is_internal());
   return InternalLayout().template Pointer<N>(reinterpret_cast<char *>(this));
 }
 template <size_type N>
 inline const typename layout_type::template ElementType<N> *GetField() const {
   assert(N < 4 || is_internal());
   return InternalLayout().template Pointer<N>(
       reinterpret_cast<const char *>(this));
 }
 void set_parent(btree_node *p) { *GetField<0>() = p; }
 field_type &mutable_finish() { return GetField<2>()[2]; }
 slot_type *slot(size_type i) { return &GetField<3>()[i]; }
 slot_type *start_slot() { return slot(start()); }
 slot_type *finish_slot() { return slot(finish()); }
 const slot_type *slot(size_type i) const { return &GetField<3>()[i]; }
 void set_position(field_type v) { GetField<2>()[0] = v; }
 void set_start(field_type v) { GetField<2>()[1] = v; }
 void set_finish(field_type v) { GetField<2>()[2] = v; }
 // This method is only called by the node init methods.
 void set_max_count(field_type v) { GetField<2>()[3] = v; }

public:
 // Whether this is a leaf node or not. This value doesn't change after the
 // node is created.
 bool is_leaf() const { return GetField<2>()[3] != kInternalNodeMaxCount; }
 // Whether this is an internal node or not. This value doesn't change after
 // the node is created.
 bool is_internal() const { return !is_leaf(); }

 // Getter for the position of this node in its parent.
 field_type position() const { return GetField<2>()[0]; }

 // Getter for the offset of the first value in the `values` array.
 field_type start() const {
   // TODO(ezb): when floating storage is implemented, return GetField<2>()[1];
   assert(GetField<2>()[1] == 0);
   return 0;
 }

 // Getter for the offset after the last value in the `values` array.
 field_type finish() const { return GetField<2>()[2]; }

 // Getters for the number of values stored in this node.
 field_type count() const {
   assert(finish() >= start());
   return finish() - start();
 }
 field_type max_count() const {
   // Internal nodes have max_count==kInternalNodeMaxCount.
   // Leaf nodes have max_count in [1, kNodeSlots].
   const field_type max_count = GetField<2>()[3];
   return max_count == field_type{kInternalNodeMaxCount}
              ? field_type{kNodeSlots}
              : max_count;
 }

 // Getter for the parent of this node.
 // TODO(ezb): assert that the child of the returned node at position
 // `node_->position()` maps to the current node.
 btree_node *parent() const { return *GetField<0>(); }
 // Getter for whether the node is the root of the tree. The parent of the
 // root of the tree is the leftmost node in the tree which is guaranteed to
 // be a leaf.
 bool is_root() const { return parent()->is_leaf(); }
 void make_root() {
   assert(parent()->is_root());
   set_generation(parent()->generation());
   set_parent(parent()->parent());
 }

 // Gets the root node's generation integer, which is the one used by the tree.
 uint32_t *get_root_generation() const {
   assert(BtreeGenerationsEnabled());
   const btree_node *curr = this;
   for (; !curr->is_root(); curr = curr->parent()) continue;
   return const_cast<uint32_t *>(&curr->GetField<1>()[0]);
 }

 // Returns the generation for iterator validation.
 uint32_t generation() const {
   return BtreeGenerationsEnabled() ? *get_root_generation() : 0;
 }
 // Updates generation. Should only be called on a root node or during node
 // initialization.
 void set_generation(uint32_t generation) {
   if (BtreeGenerationsEnabled()) GetField<1>()[0] = generation;
 }
 // Updates the generation. We do this whenever the node is mutated.
 void next_generation() {
   if (BtreeGenerationsEnabled()) ++*get_root_generation();
 }

 // Getters for the key/value at position i in the node.
 const key_type &key(size_type i) const { return params_type::key(slot(i)); }
 reference value(size_type i) { return params_type::element(slot(i)); }
 const_reference value(size_type i) const {
   return params_type::element(slot(i));
 }

 // Getters/setter for the child at position i in the node.
 btree_node *child(field_type i) const { return GetField<4>()[i]; }
 btree_node *start_child() const { return child(start()); }
 btree_node *&mutable_child(field_type i) { return GetField<4>()[i]; }
 void clear_child(field_type i) {
   absl::container_internal::SanitizerPoisonObject(&mutable_child(i));
 }
 void set_child_noupdate_position(field_type i, btree_node *c) {
   absl::container_internal::SanitizerUnpoisonObject(&mutable_child(i));
   mutable_child(i) = c;
 }
 void set_child(field_type i, btree_node *c) {
   set_child_noupdate_position(i, c);
   c->set_position(i);
 }
 void init_child(field_type i, btree_node *c) {
   set_child(i, c);
   c->set_parent(this);
 }

 // Returns the position of the first value whose key is not less than k.
 template <typename K>
 SearchResult<size_type, is_key_compare_to::value> lower_bound(
     const K &k, const key_compare &comp) const {
   return use_linear_search::value ? linear_search(k, comp)
                                   : binary_search(k, comp);
 }
 // Returns the position of the first value whose key is greater than k.
 template <typename K>
 size_type upper_bound(const K &k, const key_compare &comp) const {
   auto upper_compare = upper_bound_adapter<key_compare>(comp);
   return use_linear_search::value ? linear_search(k, upper_compare).value
                                   : binary_search(k, upper_compare).value;
 }

 template <typename K, typename Compare>
 SearchResult<size_type, btree_is_key_compare_to<Compare, key_type>::value>
 linear_search(const K &k, const Compare &comp) const {
   return linear_search_impl(k, start(), finish(), comp,
                             btree_is_key_compare_to<Compare, key_type>());
 }

 template <typename K, typename Compare>
 SearchResult<size_type, btree_is_key_compare_to<Compare, key_type>::value>
 binary_search(const K &k, const Compare &comp) const {
   return binary_search_impl(k, start(), finish(), comp,
                             btree_is_key_compare_to<Compare, key_type>());
 }

 // Returns the position of the first value whose key is not less than k using
 // linear search performed using plain compare.
 template <typename K, typename Compare>
 SearchResult<size_type, false> linear_search_impl(
     const K &k, size_type s, const size_type e, const Compare &comp,
     std::false_type /* IsCompareTo */) const {
   while (s < e) {
     if (!comp(key(s), k)) {
       break;
     }
     ++s;
   }
   return SearchResult<size_type, false>{s};
 }

 // Returns the position of the first value whose key is not less than k using
 // linear search performed using compare-to.
 template <typename K, typename Compare>
 SearchResult<size_type, true> linear_search_impl(
     const K &k, size_type s, const size_type e, const Compare &comp,
     std::true_type /* IsCompareTo */) const {
   while (s < e) {
     const absl::weak_ordering c = comp(key(s), k);
     if (c == 0) {
       return {s, MatchKind::kEq};
     } else if (c > 0) {
       break;
     }
     ++s;
   }
   return {s, MatchKind::kNe};
 }

 // Returns the position of the first value whose key is not less than k using
 // binary search performed using plain compare.
 template <typename K, typename Compare>
 SearchResult<size_type, false> binary_search_impl(
     const K &k, size_type s, size_type e, const Compare &comp,
     std::false_type /* IsCompareTo */) const {
   while (s != e) {
     const size_type mid = (s + e) >> 1;
     if (comp(key(mid), k)) {
       s = mid + 1;
     } else {
       e = mid;
     }
   }
   return SearchResult<size_type, false>{s};
 }

 // Returns the position of the first value whose key is not less than k using
 // binary search performed using compare-to.
 template <typename K, typename CompareTo>
 SearchResult<size_type, true> binary_search_impl(
     const K &k, size_type s, size_type e, const CompareTo &comp,
     std::true_type /* IsCompareTo */) const {
   if (params_type::template can_have_multiple_equivalent_keys<K>()) {
     MatchKind exact_match = MatchKind::kNe;
     while (s != e) {
       const size_type mid = (s + e) >> 1;
       const absl::weak_ordering c = comp(key(mid), k);
       if (c < 0) {
         s = mid + 1;
       } else {
         e = mid;
         if (c == 0) {
           // Need to return the first value whose key is not less than k,
           // which requires continuing the binary search if there could be
           // multiple equivalent keys.
           exact_match = MatchKind::kEq;
         }
       }
     }
     return {s, exact_match};
   } else {  // Can't have multiple equivalent keys.
     while (s != e) {
       const size_type mid = (s + e) >> 1;
       const absl::weak_ordering c = comp(key(mid), k);
       if (c < 0) {
         s = mid + 1;
       } else if (c > 0) {
         e = mid;
       } else {
         return {mid, MatchKind::kEq};
       }
     }
     return {s, MatchKind::kNe};
   }
 }

 // Returns whether key i is ordered correctly with respect to the other keys
 // in the node. The motivation here is to detect comparators that violate
 // transitivity. Note: we only do comparisons of keys on this node rather than
 // the whole tree so that this is constant time.
 template <typename Compare>
 bool is_ordered_correctly(field_type i, const Compare &comp) const {
   if (std::is_base_of<BtreeTestOnlyCheckedCompareOptOutBase,
                       Compare>::value ||
       params_type::kIsKeyCompareStringAdapted) {
     return true;
   }

   const auto compare = [&](field_type a, field_type b) {
     const absl::weak_ordering cmp =
         compare_internal::do_three_way_comparison(comp, key(a), key(b));
     return cmp < 0 ? -1 : cmp > 0 ? 1 : 0;
   };
   int cmp = -1;
   constexpr bool kCanHaveEquivKeys =
       params_type::template can_have_multiple_equivalent_keys<key_type>();
   for (field_type j = start(); j < finish(); ++j) {
     if (j == i) {
       if (cmp > 0) return false;
       continue;
     }
     int new_cmp = compare(j, i);
     if (new_cmp < cmp || (!kCanHaveEquivKeys && new_cmp == 0)) return false;
     cmp = new_cmp;
   }
   return true;
 }

 // Emplaces a value at position i, shifting all existing values and
 // children at positions >= i to the right by 1.
 template <typename... Args>
 void emplace_value(field_type i, allocator_type *alloc, Args &&...args);

 // Removes the values at positions [i, i + to_erase), shifting all existing
 // values and children after that range to the left by to_erase. Clears all
 // children between [i, i + to_erase).
 void remove_values(field_type i, field_type to_erase, allocator_type *alloc);

 // Rebalances a node with its right sibling.
 void rebalance_right_to_left(field_type to_move, btree_node *right,
                              allocator_type *alloc);
 void rebalance_left_to_right(field_type to_move, btree_node *right,
                              allocator_type *alloc);

 // Splits a node, moving a portion of the node's values to its right sibling.
 void split(int insert_position, btree_node *dest, allocator_type *alloc);

 // Merges a node with its right sibling, moving all of the values and the
 // delimiting key in the parent node onto itself, and deleting the src node.
 void merge(btree_node *src, allocator_type *alloc);

 // Node allocation/deletion routines.
 void init_leaf(field_type position, field_type max_count,
                btree_node *parent) {
   set_generation(0);
   set_parent(parent);
   set_position(position);
   set_start(0);
   set_finish(0);
   set_max_count(max_count);
   absl::container_internal::SanitizerPoisonMemoryRegion(
       start_slot(), max_count * sizeof(slot_type));
 }
 void init_internal(field_type position, btree_node *parent) {
   init_leaf(position, kNodeSlots, parent);
   // Set `max_count` to a sentinel value to indicate that this node is
   // internal.
   set_max_count(kInternalNodeMaxCount);
   absl::container_internal::SanitizerPoisonMemoryRegion(
       &mutable_child(start()), (kNodeSlots + 1) * sizeof(btree_node *));
 }

 static void deallocate(const size_type size, btree_node *node,
                        allocator_type *alloc) {
   absl::container_internal::SanitizerUnpoisonMemoryRegion(node, size);
   absl::container_internal::Deallocate<Alignment()>(alloc, node, size);
 }

 // Deletes a node and all of its children.
 static void clear_and_delete(btree_node *node, allocator_type *alloc);

private:
 template <typename... Args>
 void value_init(const field_type i, allocator_type *alloc, Args &&...args) {
   next_generation();
   absl::container_internal::SanitizerUnpoisonObject(slot(i));
   params_type::construct(alloc, slot(i), std::forward<Args>(args)...);
 }
 void value_destroy(const field_type i, allocator_type *alloc) {
   next_generation();
   params_type::destroy(alloc, slot(i));
   absl::container_internal::SanitizerPoisonObject(slot(i));
 }
 void value_destroy_n(const field_type i, const field_type n,
                      allocator_type *alloc) {
   next_generation();
   for (slot_type *s = slot(i), *end = slot(i + n); s != end; ++s) {
     params_type::destroy(alloc, s);
     absl::container_internal::SanitizerPoisonObject(s);
   }
 }

 static void transfer(slot_type *dest, slot_type *src, allocator_type *alloc) {
   absl::container_internal::SanitizerUnpoisonObject(dest);
   params_type::transfer(alloc, dest, src);
   absl::container_internal::SanitizerPoisonObject(src);
 }

 // Transfers value from slot `src_i` in `src_node` to slot `dest_i` in `this`.
 void transfer(const size_type dest_i, const size_type src_i,
               btree_node *src_node, allocator_type *alloc) {
   next_generation();
   transfer(slot(dest_i), src_node->slot(src_i), alloc);
 }

 // Transfers `n` values starting at value `src_i` in `src_node` into the
 // values starting at value `dest_i` in `this`.
 void transfer_n(const size_type n, const size_type dest_i,
                 const size_type src_i, btree_node *src_node,
                 allocator_type *alloc) {
   next_generation();
   for (slot_type *src = src_node->slot(src_i), *end = src + n,
                  *dest = slot(dest_i);
        src != end; ++src, ++dest) {
     transfer(dest, src, alloc);
   }
 }

 // Same as above, except that we start at the end and work our way to the
 // beginning.
 void transfer_n_backward(const size_type n, const size_type dest_i,
                          const size_type src_i, btree_node *src_node,
                          allocator_type *alloc) {
   next_generation();
   for (slot_type *src = src_node->slot(src_i + n), *end = src - n,
                  *dest = slot(dest_i + n);
        src != end; --src, --dest) {
     // If we modified the loop index calculations above to avoid the -1s here,
     // it would result in UB in the computation of `end` (and possibly `src`
     // as well, if n == 0), since slot() is effectively an array index and it
     // is UB to compute the address of any out-of-bounds array element except
     // for one-past-the-end.
     transfer(dest - 1, src - 1, alloc);
   }
 }

 template <typename P>
 friend class btree;
 template <typename N, typename R, typename P>
 friend class btree_iterator;
 friend class BtreeNodePeer;
 friend struct btree_access;
};

template <typename Node>
bool AreNodesFromSameContainer(const Node *node_a, const Node *node_b) {
 // If either node is null, then give up on checking whether they're from the
 // same container. (If exactly one is null, then we'll trigger the
 // default-constructed assert in Equals.)
 if (node_a == nullptr || node_b == nullptr) return true;
 while (!node_a->is_root()) node_a = node_a->parent();
 while (!node_b->is_root()) node_b = node_b->parent();
 return node_a == node_b;
}

class btree_iterator_generation_info_enabled {
public:
 explicit btree_iterator_generation_info_enabled(uint32_t g)
     : generation_(g) {}

 // Updates the generation. For use internally right before we return an
 // iterator to the user.
 template <typename Node>
 void update_generation(const Node *node) {
   if (node != nullptr) generation_ = node->generation();
 }
 uint32_t generation() const { return generation_; }

 template <typename Node>
 void assert_valid_generation(const Node *node) const {
   if (node != nullptr && node->generation() != generation_) {
     ABSL_INTERNAL_LOG(
         FATAL,
         "Attempting to use an invalidated iterator. The corresponding b-tree "
         "container has been mutated since this iterator was constructed.");
   }
 }

private:
 // Used to check that the iterator hasn't been invalidated.
 uint32_t generation_;
};

class btree_iterator_generation_info_disabled {
public:
 explicit btree_iterator_generation_info_disabled(uint32_t) {}
 static void update_generation(const void *) {}
 static uint32_t generation() { return 0; }
 static void assert_valid_generation(const void *) {}
};

#ifdef ABSL_BTREE_ENABLE_GENERATIONS
using btree_iterator_generation_info = btree_iterator_generation_info_enabled;
#else
using btree_iterator_generation_info = btree_iterator_generation_info_disabled;
#endif

template <typename Node, typename Reference, typename Pointer>
class btree_iterator : private btree_iterator_generation_info {
 using field_type = typename Node::field_type;
 using key_type = typename Node::key_type;
 using size_type = typename Node::size_type;
 using params_type = typename Node::params_type;
 using is_map_container = typename params_type::is_map_container;

 using node_type = Node;
 using normal_node = typename std::remove_const<Node>::type;
 using const_node = const Node;
 using normal_pointer = typename params_type::pointer;
 using normal_reference = typename params_type::reference;
 using const_pointer = typename params_type::const_pointer;
 using const_reference = typename params_type::const_reference;
 using slot_type = typename params_type::slot_type;

 // In sets, all iterators are const.
 using iterator = absl::conditional_t<
     is_map_container::value,
     btree_iterator<normal_node, normal_reference, normal_pointer>,
     btree_iterator<normal_node, const_reference, const_pointer>>;
 using const_iterator =
     btree_iterator<const_node, const_reference, const_pointer>;

public:
 // These aliases are public for std::iterator_traits.
 using difference_type = typename Node::difference_type;
 using value_type = typename params_type::value_type;
 using pointer = Pointer;
 using reference = Reference;
 using iterator_category = std::bidirectional_iterator_tag;

 btree_iterator() : btree_iterator(nullptr, -1) {}
 explicit btree_iterator(Node *n) : btree_iterator(n, n->start()) {}
 btree_iterator(Node *n, int p)
     : btree_iterator_generation_info(n != nullptr ? n->generation()
                                                   : ~uint32_t{}),
       node_(n),
       position_(p) {}

 // NOTE: this SFINAE allows for implicit conversions from iterator to
 // const_iterator, but it specifically avoids hiding the copy constructor so
 // that the trivial one will be used when possible.
 template <typename N, typename R, typename P,
           absl::enable_if_t<
               std::is_same<btree_iterator<N, R, P>, iterator>::value &&
                   std::is_same<btree_iterator, const_iterator>::value,
               int> = 0>
 btree_iterator(const btree_iterator<N, R, P> other)  // NOLINT
     : btree_iterator_generation_info(other),
       node_(other.node_),
       position_(other.position_) {}

 bool operator==(const iterator &other) const {
   return Equals(other);
 }
 bool operator==(const const_iterator &other) const {
   return Equals(other);
 }
 bool operator!=(const iterator &other) const {
   return !Equals(other);
 }
 bool operator!=(const const_iterator &other) const {
   return !Equals(other);
 }

 // Returns n such that n calls to ++other yields *this.
 // Precondition: n exists.
 difference_type operator-(const_iterator other) const {
   if (node_ == other.node_) {
     if (node_->is_leaf()) return position_ - other.position_;
     if (position_ == other.position_) return 0;
   }
   return distance_slow(other);
 }

 // Advances the iterator by `n`. Values of `n` must not result in going past
 // the `end` iterator (for a positive `n`) or before the `begin` iterator (for
 // a negative `n`).
 btree_iterator &operator+=(difference_type n) {
   assert_valid_generation(node_);
   if (n == 0) return *this;
   if (n < 0) return decrement_n_slow(-n);
   return increment_n_slow(n);
 }

 // Moves the iterator by `n` positions backwards. Values of `n` must not
 // result in going before the `begin` iterator (for a positive `n`) or past
 // the `end` iterator (for a negative `n`).
 btree_iterator &operator-=(difference_type n) {
   assert_valid_generation(node_);
   if (n == 0) return *this;
   if (n < 0) return increment_n_slow(-n);
   return decrement_n_slow(n);
 }

 // Accessors for the key/value the iterator is pointing at.
 reference operator*() const {
   ABSL_HARDENING_ASSERT(node_ != nullptr);
   assert_valid_generation(node_);
   ABSL_HARDENING_ASSERT(position_ >= node_->start());
   if (position_ >= node_->finish()) {
     ABSL_HARDENING_ASSERT(!IsEndIterator() && "Dereferencing end() iterator");
     ABSL_HARDENING_ASSERT(position_ < node_->finish());
   }
   return node_->value(static_cast<field_type>(position_));
 }
 pointer operator->() const { return &operator*(); }

 btree_iterator &operator++() {
   increment();
   return *this;
 }
 btree_iterator &operator--() {
   decrement();
   return *this;
 }
 btree_iterator operator++(int) {
   btree_iterator tmp = *this;
   ++*this;
   return tmp;
 }
 btree_iterator operator--(int) {
   btree_iterator tmp = *this;
   --*this;
   return tmp;
 }

private:
 friend iterator;
 friend const_iterator;
 template <typename Params>
 friend class btree;
 template <typename Tree>
 friend class btree_container;
 template <typename Tree>
 friend class btree_set_container;
 template <typename Tree>
 friend class btree_map_container;
 template <typename Tree>
 friend class btree_multiset_container;
 template <typename TreeType, typename CheckerType>
 friend class base_checker;
 friend struct btree_access;

 // This SFINAE allows explicit conversions from const_iterator to
 // iterator, but also avoids hiding the copy constructor.
 // NOTE: the const_cast is safe because this constructor is only called by
 // non-const methods and the container owns the nodes.
 template <typename N, typename R, typename P,
           absl::enable_if_t<
               std::is_same<btree_iterator<N, R, P>, const_iterator>::value &&
                   std::is_same<btree_iterator, iterator>::value,
               int> = 0>
 explicit btree_iterator(const btree_iterator<N, R, P> other)
     : btree_iterator_generation_info(other.generation()),
       node_(const_cast<node_type *>(other.node_)),
       position_(other.position_) {}

 bool Equals(const const_iterator other) const {
   ABSL_HARDENING_ASSERT(((node_ == nullptr && other.node_ == nullptr) ||
                          (node_ != nullptr && other.node_ != nullptr)) &&
                         "Comparing default-constructed iterator with "
                         "non-default-constructed iterator.");
   // Note: we use assert instead of ABSL_HARDENING_ASSERT here because this
   // changes the complexity of Equals from O(1) to O(log(N) + log(M)) where
   // N/M are sizes of the containers containing node_/other.node_.
   assert(AreNodesFromSameContainer(node_, other.node_) &&
          "Comparing iterators from different containers.");
   assert_valid_generation(node_);
   other.assert_valid_generation(other.node_);
   return node_ == other.node_ && position_ == other.position_;
 }

 bool IsEndIterator() const {
   if (position_ != node_->finish()) return false;
   node_type *node = node_;
   while (!node->is_root()) {
     if (node->position() != node->parent()->finish()) return false;
     node = node->parent();
   }
   return true;
 }

 // Returns n such that n calls to ++other yields *this.
 // Precondition: n exists && (this->node_ != other.node_ ||
 // !this->node_->is_leaf() || this->position_ != other.position_).
 difference_type distance_slow(const_iterator other) const;

 // Increment/decrement the iterator.
 void increment() {
   assert_valid_generation(node_);
   if (node_->is_leaf() && ++position_ < node_->finish()) {
     return;
   }
   increment_slow();
 }
 void increment_slow();
 btree_iterator &increment_n_slow(difference_type n);

 void decrement() {
   assert_valid_generation(node_);
   if (node_->is_leaf() && --position_ >= node_->start()) {
     return;
   }
   decrement_slow();
 }
 void decrement_slow();
 btree_iterator &decrement_n_slow(difference_type n);

 const key_type &key() const {
   return node_->key(static_cast<size_type>(position_));
 }
 decltype(std::declval<Node *>()->slot(0)) slot() {
   return node_->slot(static_cast<size_type>(position_));
 }

 void update_generation() {
   btree_iterator_generation_info::update_generation(node_);
 }

 // The node in the tree the iterator is pointing at.
 Node *node_;
 // The position within the node of the tree the iterator is pointing at.
 // NOTE: this is an int rather than a field_type because iterators can point
 // to invalid positions (such as -1) in certain circumstances.
 int position_;
};

template <typename Params>
class btree {
 using node_type = btree_node<Params>;
 using is_key_compare_to = typename Params::is_key_compare_to;
 using field_type = typename node_type::field_type;

 // We use a static empty node for the root/leftmost/rightmost of empty btrees
 // in order to avoid branching in begin()/end().
 struct EmptyNodeType : node_type {
   using field_type = typename node_type::field_type;
   node_type *parent;
#ifdef ABSL_BTREE_ENABLE_GENERATIONS
   uint32_t generation = 0;
#endif
   field_type position = 0;
   field_type start = 0;
   field_type finish = 0;
   // max_count must be != kInternalNodeMaxCount (so that this node is regarded
   // as a leaf node). max_count() is never called when the tree is empty.
   field_type max_count = node_type::kInternalNodeMaxCount + 1;

   constexpr EmptyNodeType() : parent(this) {}
 };

 static node_type *EmptyNode() {
   alignas(node_type::Alignment()) static constexpr EmptyNodeType empty_node;
   return const_cast<EmptyNodeType *>(&empty_node);
 }

 enum : uint32_t {
   kNodeSlots = node_type::kNodeSlots,
   kMinNodeValues = kNodeSlots / 2,
 };

 struct node_stats {
   using size_type = typename Params::size_type;

   node_stats(size_type l, size_type i) : leaf_nodes(l), internal_nodes(i) {}

   node_stats &operator+=(const node_stats &other) {
     leaf_nodes += other.leaf_nodes;
     internal_nodes += other.internal_nodes;
     return *this;
   }

   size_type leaf_nodes;
   size_type internal_nodes;
 };

public:
 using key_type = typename Params::key_type;
 using value_type = typename Params::value_type;
 using size_type = typename Params::size_type;
 using difference_type = typename Params::difference_type;
 using key_compare = typename Params::key_compare;
 using original_key_compare = typename Params::original_key_compare;
 using value_compare = typename Params::value_compare;
 using allocator_type = typename Params::allocator_type;
 using reference = typename Params::reference;
 using const_reference = typename Params::const_reference;
 using pointer = typename Params::pointer;
 using const_pointer = typename Params::const_pointer;
 using iterator =
     typename btree_iterator<node_type, reference, pointer>::iterator;
 using const_iterator = typename iterator::const_iterator;
 using reverse_iterator = std::reverse_iterator<iterator>;
 using const_reverse_iterator = std::reverse_iterator<const_iterator>;
 using node_handle_type = node_handle<Params, Params, allocator_type>;

 // Internal types made public for use by btree_container types.
 using params_type = Params;
 using slot_type = typename Params::slot_type;

private:
 // Copies or moves (depending on the template parameter) the values in
 // other into this btree in their order in other. This btree must be empty
 // before this method is called. This method is used in copy construction,
 // copy assignment, and move assignment.
 template <typename Btree>
 void copy_or_move_values_in_order(Btree &other);

 // Validates that various assumptions/requirements are true at compile time.
 constexpr static bool static_assert_validation();

public:
 btree(const key_compare &comp, const allocator_type &alloc)
     : root_(EmptyNode()), rightmost_(comp, alloc, EmptyNode()), size_(0) {}

 btree(const btree &other) : btree(other, other.allocator()) {}
 btree(const btree &other, const allocator_type &alloc)
     : btree(other.key_comp(), alloc) {
   copy_or_move_values_in_order(other);
 }
 btree(btree &&other) noexcept
     : root_(std::exchange(other.root_, EmptyNode())),
       rightmost_(std::move(other.rightmost_)),
       size_(std::exchange(other.size_, 0u)) {
   other.mutable_rightmost() = EmptyNode();
 }
 btree(btree &&other, const allocator_type &alloc)
     : btree(other.key_comp(), alloc) {
   if (alloc == other.allocator()) {
     swap(other);
   } else {
     // Move values from `other` one at a time when allocators are different.
     copy_or_move_values_in_order(other);
   }
 }

 ~btree() {
   // Put static_asserts in destructor to avoid triggering them before the type
   // is complete.
   static_assert(static_assert_validation(), "This call must be elided.");
   clear();
 }

 // Assign the contents of other to *this.
 btree &operator=(const btree &other);
 btree &operator=(btree &&other) noexcept;

 iterator begin() { return iterator(leftmost()); }
 const_iterator begin() const { return const_iterator(leftmost()); }
 iterator end() { return iterator(rightmost(), rightmost()->finish()); }
 const_iterator end() const {
   return const_iterator(rightmost(), rightmost()->finish());
 }
 reverse_iterator rbegin() { return reverse_iterator(end()); }
 const_reverse_iterator rbegin() const {
   return const_reverse_iterator(end());
 }
 reverse_iterator rend() { return reverse_iterator(begin()); }
 const_reverse_iterator rend() const {
   return const_reverse_iterator(begin());
 }

 // Finds the first element whose key is not less than `key`.
 template <typename K>
 iterator lower_bound(const K &key) {
   return internal_end(internal_lower_bound(key).value);
 }
 template <typename K>
 const_iterator lower_bound(const K &key) const {
   return internal_end(internal_lower_bound(key).value);
 }

 // Finds the first element whose key is not less than `key` and also returns
 // whether that element is equal to `key`.
 template <typename K>
 std::pair<iterator, bool> lower_bound_equal(const K &key) const;

 // Finds the first element whose key is greater than `key`.
 template <typename K>
 iterator upper_bound(const K &key) {
   return internal_end(internal_upper_bound(key));
 }
 template <typename K>
 const_iterator upper_bound(const K &key) const {
   return internal_end(internal_upper_bound(key));
 }

 // Finds the range of values which compare equal to key. The first member of
 // the returned pair is equal to lower_bound(key). The second member of the
 // pair is equal to upper_bound(key).
 template <typename K>
 std::pair<iterator, iterator> equal_range(const K &key);
 template <typename K>
 std::pair<const_iterator, const_iterator> equal_range(const K &key) const {
   return const_cast<btree *>(this)->equal_range(key);
 }

 // Inserts a value into the btree only if it does not already exist. The
 // boolean return value indicates whether insertion succeeded or failed.
 // Requirement: if `key` already exists in the btree, does not consume `args`.
 // Requirement: `key` is never referenced after consuming `args`.
 template <typename K, typename... Args>
 std::pair<iterator, bool> insert_unique(const K &key, Args &&...args);

 // Inserts with hint. Checks to see if the value should be placed immediately
 // before `position` in the tree. If so, then the insertion will take
 // amortized constant time. If not, the insertion will take amortized
 // logarithmic time as if a call to insert_unique() were made.
 // Requirement: if `key` already exists in the btree, does not consume `args`.
 // Requirement: `key` is never referenced after consuming `args`.
 template <typename K, typename... Args>
 std::pair<iterator, bool> insert_hint_unique(iterator position, const K &key,
                                              Args &&...args);

 // Insert a range of values into the btree.
 // Note: the first overload avoids constructing a value_type if the key
 // already exists in the btree.
 template <typename InputIterator,
           typename = decltype(std::declval<const key_compare &>()(
               params_type::key(*std::declval<InputIterator>()),
               std::declval<const key_type &>()))>
 void insert_iterator_unique(InputIterator b, InputIterator e, int);
 // We need the second overload for cases in which we need to construct a
 // value_type in order to compare it with the keys already in the btree.
 template <typename InputIterator>
 void insert_iterator_unique(InputIterator b, InputIterator e, char);

 // Inserts a value into the btree.
 template <typename ValueType>
 iterator insert_multi(const key_type &key, ValueType &&v);

 // Inserts a value into the btree.
 template <typename ValueType>
 iterator insert_multi(ValueType &&v) {
   return insert_multi(params_type::key(v), std::forward<ValueType>(v));
 }

 // Insert with hint. Check to see if the value should be placed immediately
 // before position in the tree. If it does, then the insertion will take
 // amortized constant time. If not, the insertion will take amortized
 // logarithmic time as if a call to insert_multi(v) were made.
 template <typename ValueType>
 iterator insert_hint_multi(iterator position, ValueType &&v);

 // Insert a range of values into the btree.
 template <typename InputIterator>
 void insert_iterator_multi(InputIterator b,
                            InputIterator e);

 // Erase the specified iterator from the btree. The iterator must be valid
 // (i.e. not equal to end()).  Return an iterator pointing to the node after
 // the one that was erased (or end() if none exists).
 // Requirement: does not read the value at `*iter`.
 iterator erase(iterator iter);

 // Erases range. Returns the number of keys erased and an iterator pointing
 // to the element after the last erased element.
 std::pair<size_type, iterator> erase_range(iterator begin, iterator end);

 // Finds an element with key equivalent to `key` or returns `end()` if `key`
 // is not present.
 template <typename K>
 iterator find(const K &key) {
   return internal_end(internal_find(key));
 }
 template <typename K>
 const_iterator find(const K &key) const {
   return internal_end(internal_find(key));
 }

 // Clear the btree, deleting all of the values it contains.
 void clear();

 // Swaps the contents of `this` and `other`.
 void swap(btree &other);

 const key_compare &key_comp() const noexcept {
   return rightmost_.template get<0>();
 }
 template <typename K1, typename K2>
 bool compare_keys(const K1 &a, const K2 &b) const {
   return compare_internal::compare_result_as_less_than(key_comp()(a, b));
 }

 value_compare value_comp() const {
   return value_compare(original_key_compare(key_comp()));
 }

 // Verifies the structure of the btree.
 void verify() const;

 // Size routines.
 size_type size() const { return size_; }
 size_type max_size() const { return (std::numeric_limits<size_type>::max)(); }
 bool empty() const { return size_ == 0; }

 // The height of the btree. An empty tree will have height 0.
 size_type height() const {
   size_type h = 0;
   if (!empty()) {
     // Count the length of the chain from the leftmost node up to the
     // root. We actually count from the root back around to the level below
     // the root, but the calculation is the same because of the circularity
     // of that traversal.
     const node_type *n = root();
     do {
       ++h;
       n = n->parent();
     } while (n != root());
   }
   return h;
 }

 // The number of internal, leaf and total nodes used by the btree.
 size_type leaf_nodes() const { return internal_stats(root()).leaf_nodes; }
 size_type internal_nodes() const {
   return internal_stats(root()).internal_nodes;
 }
 size_type nodes() const {
   node_stats stats = internal_stats(root());
   return stats.leaf_nodes + stats.internal_nodes;
 }

 // The total number of bytes used by the btree.
 // TODO(b/169338300): update to support node_btree_*.
 size_type bytes_used() const {
   node_stats stats = internal_stats(root());
   if (stats.leaf_nodes == 1 && stats.internal_nodes == 0) {
     return sizeof(*this) + node_type::LeafSize(root()->max_count());
   } else {
     return sizeof(*this) + stats.leaf_nodes * node_type::LeafSize() +
            stats.internal_nodes * node_type::InternalSize();
   }
 }

 // The average number of bytes used per value stored in the btree assuming
 // random insertion order.
 static double average_bytes_per_value() {
   // The expected number of values per node with random insertion order is the
   // average of the maximum and minimum numbers of values per node.
   const double expected_values_per_node = (kNodeSlots + kMinNodeValues) / 2.0;
   return node_type::LeafSize() / expected_values_per_node;
 }

 // The fullness of the btree. Computed as the number of elements in the btree
 // divided by the maximum number of elements a tree with the current number
 // of nodes could hold. A value of 1 indicates perfect space
 // utilization. Smaller values indicate space wastage.
 // Returns 0 for empty trees.
 double fullness() const {
   if (empty()) return 0.0;
   return static_cast<double>(size()) / (nodes() * kNodeSlots);
 }
 // The overhead of the btree structure in bytes per node. Computed as the
 // total number of bytes used by the btree minus the number of bytes used for
 // storing elements divided by the number of elements.
 // Returns 0 for empty trees.
 double overhead() const {
   if (empty()) return 0.0;
   return (bytes_used() - size() * sizeof(value_type)) /
          static_cast<double>(size());
 }

 // The allocator used by the btree.
 allocator_type get_allocator() const { return allocator(); }

private:
 friend struct btree_access;

 // Internal accessor routines.
 node_type *root() { return root_; }
 const node_type *root() const { return root_; }
 node_type *&mutable_root() noexcept { return root_; }
 node_type *rightmost() { return rightmost_.template get<2>(); }
 const node_type *rightmost() const { return rightmost_.template get<2>(); }
 node_type *&mutable_rightmost() noexcept {
   return rightmost_.template get<2>();
 }
 key_compare *mutable_key_comp() noexcept {
   return &rightmost_.template get<0>();
 }

 // The leftmost node is stored as the parent of the root node.
 node_type *leftmost() { return root()->parent(); }
 const node_type *leftmost() const { return root()->parent(); }

 // Allocator routines.
 allocator_type *mutable_allocator() noexcept {
   return &rightmost_.template get<1>();
 }
 const allocator_type &allocator() const noexcept {
   return rightmost_.template get<1>();
 }

 // Allocates a correctly aligned node of at least size bytes using the
 // allocator.
 node_type *allocate(size_type size) {
   return reinterpret_cast<node_type *>(
       absl::container_internal::Allocate<node_type::Alignment()>(
           mutable_allocator(), size));
 }

 // Node creation/deletion routines.
 node_type *new_internal_node(field_type position, node_type *parent) {
   node_type *n = allocate(node_type::InternalSize());
   n->init_internal(position, parent);
   return n;
 }
 node_type *new_leaf_node(field_type position, node_type *parent) {
   node_type *n = allocate(node_type::LeafSize());
   n->init_leaf(position, kNodeSlots, parent);
   return n;
 }
 node_type *new_leaf_root_node(field_type max_count) {
   node_type *n = allocate(node_type::LeafSize(max_count));
   n->init_leaf(/*position=*/0, max_count, /*parent=*/n);
   return n;
 }

 // Deletion helper routines.
 iterator rebalance_after_delete(iterator iter);

 // Rebalances or splits the node iter points to.
 void rebalance_or_split(iterator *iter);

 // Merges the values of left, right and the delimiting key on their parent
 // onto left, removing the delimiting key and deleting right.
 void merge_nodes(node_type *left, node_type *right);

 // Tries to merge node with its left or right sibling, and failing that,
 // rebalance with its left or right sibling. Returns true if a merge
 // occurred, at which point it is no longer valid to access node. Returns
 // false if no merging took place.
 bool try_merge_or_rebalance(iterator *iter);

 // Tries to shrink the height of the tree by 1.
 void try_shrink();

 iterator internal_end(iterator iter) {
   return iter.node_ != nullptr ? iter : end();
 }
 const_iterator internal_end(const_iterator iter) const {
   return iter.node_ != nullptr ? iter : end();
 }

 // Emplaces a value into the btree immediately before iter. Requires that
 // key(v) <= iter.key() and (--iter).key() <= key(v).
 template <typename... Args>
 iterator internal_emplace(iterator iter, Args &&...args);

 // Returns an iterator pointing to the first value >= the value "iter" is
 // pointing at. Note that "iter" might be pointing to an invalid location such
 // as iter.position_ == iter.node_->finish(). This routine simply moves iter
 // up in the tree to a valid location. Requires: iter.node_ is non-null.
 template <typename IterType>
 static IterType internal_last(IterType iter);

 // Returns an iterator pointing to the leaf position at which key would
 // reside in the tree, unless there is an exact match - in which case, the
 // result may not be on a leaf. When there's a three-way comparator, we can
 // return whether there was an exact match. This allows the caller to avoid a
 // subsequent comparison to determine if an exact match was made, which is
 // important for keys with expensive comparison, such as strings.
 template <typename K>
 SearchResult<iterator, is_key_compare_to::value> internal_locate(
     const K &key) const;

 // Internal routine which implements lower_bound().
 template <typename K>
 SearchResult<iterator, is_key_compare_to::value> internal_lower_bound(
     const K &key) const;

 // Internal routine which implements upper_bound().
 template <typename K>
 iterator internal_upper_bound(const K &key) const;

 // Internal routine which implements find().
 template <typename K>
 iterator internal_find(const K &key) const;

 // Verifies the tree structure of node.
 size_type internal_verify(const node_type *node, const key_type *lo,
                           const key_type *hi) const;

 node_stats internal_stats(const node_type *node) const {
   // The root can be a static empty node.
   if (node == nullptr || (node == root() && empty())) {
     return node_stats(0, 0);
   }
   if (node->is_leaf()) {
     return node_stats(1, 0);
   }
   node_stats res(0, 1);
   for (int i = node->start(); i <= node->finish(); ++i) {
     res += internal_stats(node->child(i));
   }
   return res;
 }

 node_type *root_;

 // A pointer to the rightmost node. Note that the leftmost node is stored as
 // the root's parent. We use compressed tuple in order to save space because
 // key_compare and allocator_type are usually empty.
 absl::container_internal::CompressedTuple<key_compare, allocator_type,
                                           node_type *>
     rightmost_;

 // Number of values.
 size_type size_;
};

////
// btree_node methods
template <typename P>
template <typename... Args>
inline void btree_node<P>::emplace_value(const field_type i,
                                        allocator_type *alloc,
                                        Args &&...args) {
 assert(i >= start());
 assert(i <= finish());
 // Shift old values to create space for new value and then construct it in
 // place.
 if (i < finish()) {
   transfer_n_backward(finish() - i, /*dest_i=*/i + 1, /*src_i=*/i, this,
                       alloc);
 }
 value_init(static_cast<field_type>(i), alloc, std::forward<Args>(args)...);
 set_finish(finish() + 1);

 if (is_internal() && finish() > i + 1) {
   for (field_type j = finish(); j > i + 1; --j) {
     set_child(j, child(j - 1));
   }
   clear_child(i + 1);
 }
}

template <typename P>
inline void btree_node<P>::remove_values(const field_type i,
                                        const field_type to_erase,
                                        allocator_type *alloc) {
 // Transfer values after the removed range into their new places.
 value_destroy_n(i, to_erase, alloc);
 const field_type orig_finish = finish();
 const field_type src_i = i + to_erase;
 transfer_n(orig_finish - src_i, i, src_i, this, alloc);

 if (is_internal()) {
   // Delete all children between begin and end.
   for (field_type j = 0; j < to_erase; ++j) {
     clear_and_delete(child(i + j + 1), alloc);
   }
   // Rotate children after end into new positions.
   for (field_type j = i + to_erase + 1; j <= orig_finish; ++j) {
     set_child(j - to_erase, child(j));
     clear_child(j);
   }
 }
 set_finish(orig_finish - to_erase);
}

template <typename P>
void btree_node<P>::rebalance_right_to_left(field_type to_move,
                                           btree_node *right,
                                           allocator_type *alloc) {
 assert(parent() == right->parent());
 assert(position() + 1 == right->position());
 assert(right->count() >= count());
 assert(to_move >= 1);
 assert(to_move <= right->count());

 // 1) Move the delimiting value in the parent to the left node.
 transfer(finish(), position(), parent(), alloc);

 // 2) Move the (to_move - 1) values from the right node to the left node.
 transfer_n(to_move - 1, finish() + 1, right->start(), right, alloc);

 // 3) Move the new delimiting value to the parent from the right node.
 parent()->transfer(position(), right->start() + to_move - 1, right, alloc);

 // 4) Shift the values in the right node to their correct positions.
 right->transfer_n(right->count() - to_move, right->start(),
                   right->start() + to_move, right, alloc);

 if (is_internal()) {
   // Move the child pointers from the right to the left node.
   for (field_type i = 0; i < to_move; ++i) {
     init_child(finish() + i + 1, right->child(i));
   }
   for (field_type i = right->start(); i <= right->finish() - to_move; ++i) {
     assert(i + to_move <= right->max_count());
     right->init_child(i, right->child(i + to_move));
     right->clear_child(i + to_move);
   }
 }

 // Fixup `finish` on the left and right nodes.
 set_finish(finish() + to_move);
 right->set_finish(right->finish() - to_move);
}

template <typename P>
void btree_node<P>::rebalance_left_to_right(field_type to_move,
                                           btree_node *right,
                                           allocator_type *alloc) {
 assert(parent() == right->parent());
 assert(position() + 1 == right->position());
 assert(count() >= right->count());
 assert(to_move >= 1);
 assert(to_move <= count());

 // Values in the right node are shifted to the right to make room for the
 // new to_move values. Then, the delimiting value in the parent and the
 // other (to_move - 1) values in the left node are moved into the right node.
 // Lastly, a new delimiting value is moved from the left node into the
 // parent, and the remaining empty left node entries are destroyed.

 // 1) Shift existing values in the right node to their correct positions.
 right->transfer_n_backward(right->count(), right->start() + to_move,
                            right->start(), right, alloc);

 // 2) Move the delimiting value in the parent to the right node.
 right->transfer(right->start() + to_move - 1, position(), parent(), alloc);

 // 3) Move the (to_move - 1) values from the left node to the right node.
 right->transfer_n(to_move - 1, right->start(), finish() - (to_move - 1), this,
                   alloc);

 // 4) Move the new delimiting value to the parent from the left node.
 parent()->transfer(position(), finish() - to_move, this, alloc);

 if (is_internal()) {
   // Move the child pointers from the left to the right node.
   for (field_type i = right->finish() + 1; i > right->start(); --i) {
     right->init_child(i - 1 + to_move, right->child(i - 1));
     right->clear_child(i - 1);
   }
   for (field_type i = 1; i <= to_move; ++i) {
     right->init_child(i - 1, child(finish() - to_move + i));
     clear_child(finish() - to_move + i);
   }
 }

 // Fixup the counts on the left and right nodes.
 set_finish(finish() - to_move);
 right->set_finish(right->finish() + to_move);
}

template <typename P>
void btree_node<P>::split(const int insert_position, btree_node *dest,
                         allocator_type *alloc) {
 assert(dest->count() == 0);
 assert(max_count() == kNodeSlots);
 assert(position() + 1 == dest->position());
 assert(parent() == dest->parent());

 // We bias the split based on the position being inserted. If we're
 // inserting at the beginning of the left node then bias the split to put
 // more values on the right node. If we're inserting at the end of the
 // right node then bias the split to put more values on the left node.
 if (insert_position == start()) {
   dest->set_finish(dest->start() + finish() - 1);
 } else if (insert_position == kNodeSlots) {
   dest->set_finish(dest->start());
 } else {
   dest->set_finish(dest->start() + count() / 2);
 }
 set_finish(finish() - dest->count());
 assert(count() >= 1);

 // Move values from the left sibling to the right sibling.
 dest->transfer_n(dest->count(), dest->start(), finish(), this, alloc);

 // The split key is the largest value in the left sibling.
 --mutable_finish();
 parent()->emplace_value(position(), alloc, finish_slot());
 value_destroy(finish(), alloc);
 parent()->set_child_noupdate_position(position() + 1, dest);

 if (is_internal()) {
   for (field_type i = dest->start(), j = finish() + 1; i <= dest->finish();
        ++i, ++j) {
     assert(child(j) != nullptr);
     dest->init_child(i, child(j));
     clear_child(j);
   }
 }
}

template <typename P>
void btree_node<P>::merge(btree_node *src, allocator_type *alloc) {
 assert(parent() == src->parent());
 assert(position() + 1 == src->position());

 // Move the delimiting value to the left node.
 value_init(finish(), alloc, parent()->slot(position()));

 // Move the values from the right to the left node.
 transfer_n(src->count(), finish() + 1, src->start(), src, alloc);

 if (is_internal()) {
   // Move the child pointers from the right to the left node.
   for (field_type i = src->start(), j = finish() + 1; i <= src->finish();
        ++i, ++j) {
     init_child(j, src->child(i));
     src->clear_child(i);
   }
 }

 // Fixup `finish` on the src and dest nodes.
 set_finish(start() + 1 + count() + src->count());
 src->set_finish(src->start());

 // Remove the value on the parent node and delete the src node.
 parent()->remove_values(position(), /*to_erase=*/1, alloc);
}

template <typename P>
void btree_node<P>::clear_and_delete(btree_node *node, allocator_type *alloc) {
 if (node->is_leaf()) {
   node->value_destroy_n(node->start(), node->count(), alloc);
   deallocate(LeafSize(node->max_count()), node, alloc);
   return;
 }
 if (node->count() == 0) {
   deallocate(InternalSize(), node, alloc);
   return;
 }

 // The parent of the root of the subtree we are deleting.
 btree_node *delete_root_parent = node->parent();

 // Navigate to the leftmost leaf under node, and then delete upwards.
 while (node->is_internal()) node = node->start_child();
#ifdef ABSL_BTREE_ENABLE_GENERATIONS
 // When generations are enabled, we delete the leftmost leaf last in case it's
 // the parent of the root and we need to check whether it's a leaf before we
 // can update the root's generation.
 // TODO(ezb): if we change btree_node::is_root to check a bool inside the node
 // instead of checking whether the parent is a leaf, we can remove this logic.
 btree_node *leftmost_leaf = node;
#endif
 // Use `size_type` because `pos` needs to be able to hold `kNodeSlots+1`,
 // which isn't guaranteed to be a valid `field_type`.
 size_type pos = node->position();
 btree_node *parent = node->parent();
 for (;;) {
   // In each iteration of the next loop, we delete one leaf node and go right.
   assert(pos <= parent->finish());
   do {
     node = parent->child(static_cast<field_type>(pos));
     if (node->is_internal()) {
       // Navigate to the leftmost leaf under node.
       while (node->is_internal()) node = node->start_child();
       pos = node->position();
       parent = node->parent();
     }
     node->value_destroy_n(node->start(), node->count(), alloc);
#ifdef ABSL_BTREE_ENABLE_GENERATIONS
     if (leftmost_leaf != node)
#endif
       deallocate(LeafSize(node->max_count()), node, alloc);
     ++pos;
   } while (pos <= parent->finish());

   // Once we've deleted all children of parent, delete parent and go up/right.
   assert(pos > parent->finish());
   do {
     node = parent;
     pos = node->position();
     parent = node->parent();
     node->value_destroy_n(node->start(), node->count(), alloc);
     deallocate(InternalSize(), node, alloc);
     if (parent == delete_root_parent) {
#ifdef ABSL_BTREE_ENABLE_GENERATIONS
       deallocate(LeafSize(leftmost_leaf->max_count()), leftmost_leaf, alloc);
#endif
       return;
     }
     ++pos;
   } while (pos > parent->finish());
 }
}

////
// btree_iterator methods

// Note: the implementation here is based on btree_node::clear_and_delete.
template <typename N, typename R, typename P>
auto btree_iterator<N, R, P>::distance_slow(const_iterator other) const
   -> difference_type {
 const_iterator begin = other;
 const_iterator end = *this;
 assert(begin.node_ != end.node_ || !begin.node_->is_leaf() ||
        begin.position_ != end.position_);

 const node_type *node = begin.node_;
 // We need to compensate for double counting if begin.node_ is a leaf node.
 difference_type count = node->is_leaf() ? -begin.position_ : 0;

 // First navigate to the leftmost leaf node past begin.
 if (node->is_internal()) {
   ++count;
   node = node->child(begin.position_ + 1);
 }
 while (node->is_internal()) node = node->start_child();

 // Use `size_type` because `pos` needs to be able to hold `kNodeSlots+1`,
 // which isn't guaranteed to be a valid `field_type`.
 size_type pos = node->position();
 const node_type *parent = node->parent();
 for (;;) {
   // In each iteration of the next loop, we count one leaf node and go right.
   assert(pos <= parent->finish());
   do {
     node = parent->child(static_cast<field_type>(pos));
     if (node->is_internal()) {
       // Navigate to the leftmost leaf under node.
       while (node->is_internal()) node = node->start_child();
       pos = node->position();
       parent = node->parent();
     }
     if (node == end.node_) return count + end.position_;
     if (parent == end.node_ && pos == static_cast<size_type>(end.position_))
       return count + node->count();
     // +1 is for the next internal node value.
     count += node->count() + 1;
     ++pos;
   } while (pos <= parent->finish());

   // Once we've counted all children of parent, go up/right.
   assert(pos > parent->finish());
   do {
     node = parent;
     pos = node->position();
     parent = node->parent();
     // -1 because we counted the value at end and shouldn't.
     if (parent == end.node_ && pos == static_cast<size_type>(end.position_))
       return count - 1;
     ++pos;
   } while (pos > parent->finish());
 }
}

template <typename N, typename R, typename P>
void btree_iterator<N, R, P>::increment_slow() {
 N* node = node_;
 int position = position_;
 if (node->is_leaf()) {
   assert(position >= node->finish());
   while (position == node->finish() && !node->is_root()) {
     assert(node->parent()->child(node->position()) == node);
     position = node->position();
     node = node->parent();
   }
   // TODO(ezb): assert we aren't incrementing end() instead of handling.
   if (position == node->finish()) {
     return;
   }
 } else {
   assert(position < node->finish());
   node = node->child(static_cast<field_type>(position + 1));
   while (node->is_internal()) {
     node = node->start_child();
   }
   position = node->start();
 }
 *this = {node, position};
}

template <typename N, typename R, typename P>
void btree_iterator<N, R, P>::decrement_slow() {
 N* node = node_;
 int position = position_;
 if (node->is_leaf()) {
   assert(position <= -1);
   while (position < node->start() && !node->is_root()) {
     assert(node->parent()->child(node->position()) == node);
     position = node->position() - 1;
     node = node->parent();
   }
   // TODO(ezb): assert we aren't decrementing begin() instead of handling.
   if (position < node->start()) {
     return;
   }
 } else {
   assert(position >= node->start());
   node = node->child(static_cast<field_type>(position));
   while (node->is_internal()) {
     node = node->child(node->finish());
   }
   position = node->finish() - 1;
 }
 *this = {node, position};
}

template <typename N, typename R, typename P>
btree_iterator<N, R, P> &btree_iterator<N, R, P>::increment_n_slow(
   difference_type n) {
 N *node = node_;
 int position = position_;
 ABSL_ASSUME(n > 0);
 while (n > 0) {
   if (node->is_leaf()) {
     if (position + n < node->finish()) {
       position += n;
       break;
     } else {
       n -= node->finish() - position;
       position = node->finish();
       btree_iterator save = {node, position};
       while (position == node->finish() && !node->is_root()) {
         position = node->position();
         node = node->parent();
       }
       if (position == node->finish()) {
         ABSL_HARDENING_ASSERT(n == 0);
         return *this = save;
       }
     }
   } else {
     --n;
     assert(position < node->finish());
     node = node->child(static_cast<field_type>(position + 1));
     while (node->is_internal()) {
       node = node->start_child();
     }
     position = node->start();
   }
 }
 node_ = node;
 position_ = position;
 return *this;
}

template <typename N, typename R, typename P>
btree_iterator<N, R, P> &btree_iterator<N, R, P>::decrement_n_slow(
   difference_type n) {
 N *node = node_;
 int position = position_;
 ABSL_ASSUME(n > 0);
 while (n > 0) {
   if (node->is_leaf()) {
     if (position - n >= node->start()) {
       position -= n;
       break;
     } else {
       n -= 1 + position - node->start();
       position = node->start() - 1;
       while (position < node->start() && !node->is_root()) {
         position = node->position() - 1;
         node = node->parent();
       }
       ABSL_HARDENING_ASSERT(position >= node->start());
     }
   } else {
     --n;
     assert(position >= node->start());
     node = node->child(static_cast<field_type>(position));
     while (node->is_internal()) {
       node = node->child(node->finish());
     }
     position = node->finish() - 1;
   }
 }
 node_ = node;
 position_ = position;
 return *this;
}

////
// btree methods
template <typename P>
template <typename Btree>
void btree<P>::copy_or_move_values_in_order(Btree &other) {
 static_assert(std::is_same<btree, Btree>::value ||
                   std::is_same<const btree, Btree>::value,
               "Btree type must be same or const.");
 assert(empty());

 // We can avoid key comparisons because we know the order of the
 // values is the same order we'll store them in.
 auto iter = other.begin();
 if (iter == other.end()) return;
 insert_multi(iter.slot());
 ++iter;
 for (; iter != other.end(); ++iter) {
   // If the btree is not empty, we can just insert the new value at the end
   // of the tree.
   internal_emplace(end(), iter.slot());
 }
}

template <typename P>
constexpr bool btree<P>::static_assert_validation() {
 static_assert(std::is_nothrow_copy_constructible<key_compare>::value,
               "Key comparison must be nothrow copy constructible");
 static_assert(std::is_nothrow_copy_constructible<allocator_type>::value,
               "Allocator must be nothrow copy constructible");
 static_assert(std::is_trivially_copyable<iterator>::value,
               "iterator not trivially copyable.");

 // Note: We assert that kTargetValues, which is computed from
 // Params::kTargetNodeSize, must fit the node_type::field_type.
 static_assert(
     kNodeSlots < (1 << (8 * sizeof(typename node_type::field_type))),
     "target node size too large");

 // Verify that key_compare returns an absl::{weak,strong}_ordering or bool.
 static_assert(
     compare_has_valid_result_type<key_compare, key_type>(),
     "key comparison function must return absl::{weak,strong}_ordering or "
     "bool.");

 // Test the assumption made in setting kNodeSlotSpace.
 static_assert(node_type::MinimumOverhead() >= sizeof(void *) + 4,
               "node space assumption incorrect");

 return true;
}

template <typename P>
template <typename K>
auto btree<P>::lower_bound_equal(const K &key) const
   -> std::pair<iterator, bool> {
 const SearchResult<iterator, is_key_compare_to::value> res =
     internal_lower_bound(key);
 const iterator lower = iterator(internal_end(res.value));
 const bool equal = res.HasMatch()
                        ? res.IsEq()
                        : lower != end() && !compare_keys(key, lower.key());
 return {lower, equal};
}

template <typename P>
template <typename K>
auto btree<P>::equal_range(const K &key) -> std::pair<iterator, iterator> {
 const std::pair<iterator, bool> lower_and_equal = lower_bound_equal(key);
 const iterator lower = lower_and_equal.first;
 if (!lower_and_equal.second) {
   return {lower, lower};
 }

 const iterator next = std::next(lower);
 if (!params_type::template can_have_multiple_equivalent_keys<K>()) {
   // The next iterator after lower must point to a key greater than `key`.
   // Note: if this assert fails, then it may indicate that the comparator does
   // not meet the equivalence requirements for Compare
   // (see https://en.cppreference.com/w/cpp/named_req/Compare).
   assert(next == end() || compare_keys(key, next.key()));
   return {lower, next};
 }
 // Try once more to avoid the call to upper_bound() if there's only one
 // equivalent key. This should prevent all calls to upper_bound() in cases of
 // unique-containers with heterogeneous comparators in which all comparison
 // operators have the same equivalence classes.
 if (next == end() || compare_keys(key, next.key())) return {lower, next};

 // In this case, we need to call upper_bound() to avoid worst case O(N)
 // behavior if we were to iterate over equal keys.
 return {lower, upper_bound(key)};
}

template <typename P>
template <typename K, typename... Args>
auto btree<P>::insert_unique(const K &key, Args &&...args)
   -> std::pair<iterator, bool> {
 if (empty()) {
   mutable_root() = mutable_rightmost() = new_leaf_root_node(1);
 }

 SearchResult<iterator, is_key_compare_to::value> res = internal_locate(key);
 iterator iter = res.value;

 if (res.HasMatch()) {
   if (res.IsEq()) {
     // The key already exists in the tree, do nothing.
     return {iter, false};
   }
 } else {
   iterator last = internal_last(iter);
   if (last.node_ && !compare_keys(key, last.key())) {
     // The key already exists in the tree, do nothing.
     return {last, false};
   }
 }
 return {internal_emplace(iter, std::forward<Args>(args)...), true};
}

template <typename P>
template <typename K, typename... Args>
inline auto btree<P>::insert_hint_unique(iterator position, const K &key,
                                        Args &&...args)
   -> std::pair<iterator, bool> {
 if (!empty()) {
   if (position == end() || compare_keys(key, position.key())) {
     if (position == begin() || compare_keys(std::prev(position).key(), key)) {
       // prev.key() < key < position.key()
       return {internal_emplace(position, std::forward<Args>(args)...), true};
     }
   } else if (compare_keys(position.key(), key)) {
     ++position;
     if (position == end() || compare_keys(key, position.key())) {
       // {original `position`}.key() < key < {current `position`}.key()
       return {internal_emplace(position, std::forward<Args>(args)...), true};
     }
   } else {
     // position.key() == key
     return {position, false};
   }
 }
 return insert_unique(key, std::forward<Args>(args)...);
}

template <typename P>
template <typename InputIterator, typename>
void btree<P>::insert_iterator_unique(InputIterator b, InputIterator e, int) {
 for (; b != e; ++b) {
   insert_hint_unique(end(), params_type::key(*b), *b);
 }
}

template <typename P>
template <typename InputIterator>
void btree<P>::insert_iterator_unique(InputIterator b, InputIterator e, char) {
 for (; b != e; ++b) {
   // Use a node handle to manage a temp slot.
   auto node_handle =
       CommonAccess::Construct<node_handle_type>(get_allocator(), *b);
   slot_type *slot = CommonAccess::GetSlot(node_handle);
   insert_hint_unique(end(), params_type::key(slot), slot);
 }
}

template <typename P>
template <typename ValueType>
auto btree<P>::insert_multi(const key_type &key, ValueType &&v) -> iterator {
 if (empty()) {
   mutable_root() = mutable_rightmost() = new_leaf_root_node(1);
 }

 iterator iter = internal_upper_bound(key);
 if (iter.node_ == nullptr) {
   iter = end();
 }
 return internal_emplace(iter, std::forward<ValueType>(v));
}

template <typename P>
template <typename ValueType>
auto btree<P>::insert_hint_multi(iterator position, ValueType &&v) -> iterator {
 if (!empty()) {
   const key_type &key = params_type::key(v);
   if (position == end() || !compare_keys(position.key(), key)) {
     if (position == begin() ||
         !compare_keys(key, std::prev(position).key())) {
       // prev.key() <= key <= position.key()
       return internal_emplace(position, std::forward<ValueType>(v));
     }
   } else {
     ++position;
     if (position == end() || !compare_keys(position.key(), key)) {
       // {original `position`}.key() < key < {current `position`}.key()
       return internal_emplace(position, std::forward<ValueType>(v));
     }
   }
 }
 return insert_multi(std::forward<ValueType>(v));
}

template <typename P>
template <typename InputIterator>
void btree<P>::insert_iterator_multi(InputIterator b, InputIterator e) {
 for (; b != e; ++b) {
   insert_hint_multi(end(), *b);
 }
}

template <typename P>
auto btree<P>::operator=(const btree &other) -> btree & {
 if (this != &other) {
   clear();

   *mutable_key_comp() = other.key_comp();
   if (absl::allocator_traits<
           allocator_type>::propagate_on_container_copy_assignment::value) {
     *mutable_allocator() = other.allocator();
   }

   copy_or_move_values_in_order(other);
 }
 return *this;
}

template <typename P>
auto btree<P>::operator=(btree &&other) noexcept -> btree & {
 if (this != &other) {
   clear();

   using std::swap;
   if (absl::allocator_traits<
           allocator_type>::propagate_on_container_move_assignment::value) {
     swap(root_, other.root_);
     // Note: `rightmost_` also contains the allocator and the key comparator.
     swap(rightmost_, other.rightmost_);
     swap(size_, other.size_);
   } else {
     if (allocator() == other.allocator()) {
       swap(mutable_root(), other.mutable_root());
       swap(*mutable_key_comp(), *other.mutable_key_comp());
       swap(mutable_rightmost(), other.mutable_rightmost());
       swap(size_, other.size_);
     } else {
       // We aren't allowed to propagate the allocator and the allocator is
       // different so we can't take over its memory. We must move each element
       // individually. We need both `other` and `this` to have `other`s key
       // comparator while moving the values so we can't swap the key
       // comparators.
       *mutable_key_comp() = other.key_comp();
       copy_or_move_values_in_order(other);
     }
   }
 }
 return *this;
}

template <typename P>
auto btree<P>::erase(iterator iter) -> iterator {
 iter.node_->value_destroy(static_cast<field_type>(iter.position_),
                           mutable_allocator());
 iter.update_generation();

 const bool internal_delete = iter.node_->is_internal();
 if (internal_delete) {
   // Deletion of a value on an internal node. First, transfer the largest
   // value from our left child here, then erase/rebalance from that position.
   // We can get to the largest value from our left child by decrementing iter.
   iterator internal_iter(iter);
   --iter;
   assert(iter.node_->is_leaf());
   internal_iter.node_->transfer(
       static_cast<size_type>(internal_iter.position_),
       static_cast<size_type>(iter.position_), iter.node_,
       mutable_allocator());
 } else {
   // Shift values after erased position in leaf. In the internal case, we
   // don't need to do this because the leaf position is the end of the node.
   const field_type transfer_from =
       static_cast<field_type>(iter.position_ + 1);
   const field_type num_to_transfer = iter.node_->finish() - transfer_from;
   iter.node_->transfer_n(num_to_transfer,
                          static_cast<size_type>(iter.position_),
                          transfer_from, iter.node_, mutable_allocator());
 }
 // Update node finish and container size.
 iter.node_->set_finish(iter.node_->finish() - 1);
 --size_;

 // We want to return the next value after the one we just erased. If we
 // erased from an internal node (internal_delete == true), then the next
 // value is ++(++iter). If we erased from a leaf node (internal_delete ==
 // false) then the next value is ++iter. Note that ++iter may point to an
 // internal node and the value in the internal node may move to a leaf node
 // (iter.node_) when rebalancing is performed at the leaf level.

 iterator res = rebalance_after_delete(iter);

 // If we erased from an internal node, advance the iterator.
 if (internal_delete) {
   ++res;
 }
 return res;
}

template <typename P>
auto btree<P>::rebalance_after_delete(iterator iter) -> iterator {
 // Merge/rebalance as we walk back up the tree.
 iterator res(iter);
 bool first_iteration = true;
 for (;;) {
   if (iter.node_ == root()) {
     try_shrink();
     if (empty()) {
       return end();
     }
     break;
   }
   if (iter.node_->count() >= kMinNodeValues) {
     break;
   }
   bool merged = try_merge_or_rebalance(&iter);
   // On the first iteration, we should update `res` with `iter` because `res`
   // may have been invalidated.
   if (first_iteration) {
     res = iter;
     first_iteration = false;
   }
   if (!merged) {
     break;
   }
   iter.position_ = iter.node_->position();
   iter.node_ = iter.node_->parent();
 }
 res.update_generation();

 // Adjust our return value. If we're pointing at the end of a node, advance
 // the iterator.
 if (res.position_ == res.node_->finish()) {
   res.position_ = res.node_->finish() - 1;
   ++res;
 }

 return res;
}

// Note: we tried implementing this more efficiently by erasing all of the
// elements in [begin, end) at once and then doing rebalancing once at the end
// (rather than interleaving deletion and rebalancing), but that adds a lot of
// complexity, which seems to outweigh the performance win.
template <typename P>
auto btree<P>::erase_range(iterator begin, iterator end)
   -> std::pair<size_type, iterator> {
 size_type count = static_cast<size_type>(end - begin);
 assert(count >= 0);

 if (count == 0) {
   return {0, begin};
 }

 if (static_cast<size_type>(count) == size_) {
   clear();
   return {count, this->end()};
 }

 if (begin.node_ == end.node_) {
   assert(end.position_ > begin.position_);
   begin.node_->remove_values(
       static_cast<field_type>(begin.position_),
       static_cast<field_type>(end.position_ - begin.position_),
       mutable_allocator());
   size_ -= count;
   return {count, rebalance_after_delete(begin)};
 }

 const size_type target_size = size_ - count;
 while (size_ > target_size) {
   if (begin.node_->is_leaf()) {
     const size_type remaining_to_erase = size_ - target_size;
     const size_type remaining_in_node =
         static_cast<size_type>(begin.node_->finish() - begin.position_);
     const field_type to_erase = static_cast<field_type>(
         (std::min)(remaining_to_erase, remaining_in_node));
     begin.node_->remove_values(static_cast<field_type>(begin.position_),
                                to_erase, mutable_allocator());
     size_ -= to_erase;
     begin = rebalance_after_delete(begin);
   } else {
     begin = erase(begin);
   }
 }
 begin.update_generation();
 return {count, begin};
}

template <typename P>
void btree<P>::clear() {
 if (!empty()) {
   node_type::clear_and_delete(root(), mutable_allocator());
 }
 mutable_root() = mutable_rightmost() = EmptyNode();
 size_ = 0;
}

template <typename P>
void btree<P>::swap(btree &other) {
 using std::swap;
 if (absl::allocator_traits<
         allocator_type>::propagate_on_container_swap::value) {
   // Note: `rightmost_` also contains the allocator and the key comparator.
   swap(rightmost_, other.rightmost_);
 } else {
   // It's undefined behavior if the allocators are unequal here.
   assert(allocator() == other.allocator());
   swap(mutable_rightmost(), other.mutable_rightmost());
   swap(*mutable_key_comp(), *other.mutable_key_comp());
 }
 swap(mutable_root(), other.mutable_root());
 swap(size_, other.size_);
}

template <typename P>
void btree<P>::verify() const {
 assert(root() != nullptr);
 assert(leftmost() != nullptr);
 assert(rightmost() != nullptr);
 assert(empty() || size() == internal_verify(root(), nullptr, nullptr));
 assert(leftmost() == (++const_iterator(root(), -1)).node_);
 assert(rightmost() == (--const_iterator(root(), root()->finish())).node_);
 assert(leftmost()->is_leaf());
 assert(rightmost()->is_leaf());
}

template <typename P>
void btree<P>::rebalance_or_split(iterator *iter) {
 node_type *&node = iter->node_;
 int &insert_position = iter->position_;
 assert(node->count() == node->max_count());
 assert(kNodeSlots == node->max_count());

 // First try to make room on the node by rebalancing.
 node_type *parent = node->parent();
 if (node != root()) {
   if (node->position() > parent->start()) {
     // Try rebalancing with our left sibling.
     node_type *left = parent->child(node->position() - 1);
     assert(left->max_count() == kNodeSlots);
     if (left->count() < kNodeSlots) {
       // We bias rebalancing based on the position being inserted. If we're
       // inserting at the end of the right node then we bias rebalancing to
       // fill up the left node.
       field_type to_move =
           (kNodeSlots - left->count()) /
           (1 + (static_cast<field_type>(insert_position) < kNodeSlots));
       to_move = (std::max)(field_type{1}, to_move);

       if (static_cast<field_type>(insert_position) - to_move >=
               node->start() ||
           left->count() + to_move < kNodeSlots) {
         left->rebalance_right_to_left(to_move, node, mutable_allocator());

         assert(node->max_count() - node->count() == to_move);
         insert_position = static_cast<int>(
             static_cast<field_type>(insert_position) - to_move);
         if (insert_position < node->start()) {
           insert_position = insert_position + left->count() + 1;
           node = left;
         }

         assert(node->count() < node->max_count());
         return;
       }
     }
   }

   if (node->position() < parent->finish()) {
     // Try rebalancing with our right sibling.
     node_type *right = parent->child(node->position() + 1);
     assert(right->max_count() == kNodeSlots);
     if (right->count() < kNodeSlots) {
       // We bias rebalancing based on the position being inserted. If we're
       // inserting at the beginning of the left node then we bias rebalancing
       // to fill up the right node.
       field_type to_move = (kNodeSlots - right->count()) /
                            (1 + (insert_position > node->start()));
       to_move = (std::max)(field_type{1}, to_move);

       if (static_cast<field_type>(insert_position) <=
               node->finish() - to_move ||
           right->count() + to_move < kNodeSlots) {
         node->rebalance_left_to_right(to_move, right, mutable_allocator());

         if (insert_position > node->finish()) {
           insert_position = insert_position - node->count() - 1;
           node = right;
         }

         assert(node->count() < node->max_count());
         return;
       }
     }
   }

   // Rebalancing failed, make sure there is room on the parent node for a new
   // value.
   assert(parent->max_count() == kNodeSlots);
   if (parent->count() == kNodeSlots) {
     iterator parent_iter(parent, node->position());
     rebalance_or_split(&parent_iter);
     parent = node->parent();
   }
 } else {
   // Rebalancing not possible because this is the root node.
   // Create a new root node and set the current root node as the child of the
   // new root.
   parent = new_internal_node(/*position=*/0, parent);
   parent->set_generation(root()->generation());
   parent->init_child(parent->start(), node);
   mutable_root() = parent;
   // If the former root was a leaf node, then it's now the rightmost node.
   assert(parent->start_child()->is_internal() ||
          parent->start_child() == rightmost());
 }

 // Split the node.
 node_type *split_node;
 if (node->is_leaf()) {
   split_node = new_leaf_node(node->position() + 1, parent);
   node->split(insert_position, split_node, mutable_allocator());
   if (rightmost() == node) mutable_rightmost() = split_node;
 } else {
   split_node = new_internal_node(node->position() + 1, parent);
   node->split(insert_position, split_node, mutable_allocator());
 }

 if (insert_position > node->finish()) {
   insert_position = insert_position - node->count() - 1;
   node = split_node;
 }
}

template <typename P>
void btree<P>::merge_nodes(node_type *left, node_type *right) {
 left->merge(right, mutable_allocator());
 if (rightmost() == right) mutable_rightmost() = left;
}

template <typename P>
bool btree<P>::try_merge_or_rebalance(iterator *iter) {
 node_type *parent = iter->node_->parent();
 if (iter->node_->position() > parent->start()) {
   // Try merging with our left sibling.
   node_type *left = parent->child(iter->node_->position() - 1);
   assert(left->max_count() == kNodeSlots);
   if (1U + left->count() + iter->node_->count() <= kNodeSlots) {
     iter->position_ += 1 + left->count();
     merge_nodes(left, iter->node_);
     iter->node_ = left;
     return true;
   }
 }
 if (iter->node_->position() < parent->finish()) {
   // Try merging with our right sibling.
   node_type *right = parent->child(iter->node_->position() + 1);
   assert(right->max_count() == kNodeSlots);
   if (1U + iter->node_->count() + right->count() <= kNodeSlots) {
     merge_nodes(iter->node_, right);
     return true;
   }
   // Try rebalancing with our right sibling. We don't perform rebalancing if
   // we deleted the first element from iter->node_ and the node is not
   // empty. This is a small optimization for the common pattern of deleting
   // from the front of the tree.
   if (right->count() > kMinNodeValues &&
       (iter->node_->count() == 0 || iter->position_ > iter->node_->start())) {
     field_type to_move = (right->count() - iter->node_->count()) / 2;
     to_move =
         (std::min)(to_move, static_cast<field_type>(right->count() - 1));
     iter->node_->rebalance_right_to_left(to_move, right, mutable_allocator());
     return false;
   }
 }
 if (iter->node_->position() > parent->start()) {
   // Try rebalancing with our left sibling. We don't perform rebalancing if
   // we deleted the last element from iter->node_ and the node is not
   // empty. This is a small optimization for the common pattern of deleting
   // from the back of the tree.
   node_type *left = parent->child(iter->node_->position() - 1);
   if (left->count() > kMinNodeValues &&
       (iter->node_->count() == 0 ||
        iter->position_ < iter->node_->finish())) {
     field_type to_move = (left->count() - iter->node_->count()) / 2;
     to_move = (std::min)(to_move, static_cast<field_type>(left->count() - 1));
     left->rebalance_left_to_right(to_move, iter->node_, mutable_allocator());
     iter->position_ += to_move;
     return false;
   }
 }
 return false;
}

template <typename P>
void btree<P>::try_shrink() {
 node_type *orig_root = root();
 if (orig_root->count() > 0) {
   return;
 }
 // Deleted the last item on the root node, shrink the height of the tree.
 if (orig_root->is_leaf()) {
   assert(size() == 0);
   mutable_root() = mutable_rightmost() = EmptyNode();
 } else {
   node_type *child = orig_root->start_child();
   child->make_root();
   mutable_root() = child;
 }
 node_type::clear_and_delete(orig_root, mutable_allocator());
}

template <typename P>
template <typename IterType>
inline IterType btree<P>::internal_last(IterType iter) {
 assert(iter.node_ != nullptr);
 while (iter.position_ == iter.node_->finish()) {
   iter.position_ = iter.node_->position();
   iter.node_ = iter.node_->parent();
   if (iter.node_->is_leaf()) {
     iter.node_ = nullptr;
     break;
   }
 }
 iter.update_generation();
 return iter;
}

template <typename P>
template <typename... Args>
inline auto btree<P>::internal_emplace(iterator iter, Args &&...args)
   -> iterator {
 if (iter.node_->is_internal()) {
   // We can't insert on an internal node. Instead, we'll insert after the
   // previous value which is guaranteed to be on a leaf node.
   --iter;
   ++iter.position_;
 }
 const field_type max_count = iter.node_->max_count();
 allocator_type *alloc = mutable_allocator();

 const auto transfer_and_delete = [&](node_type *old_node,
                                      node_type *new_node) {
   new_node->transfer_n(old_node->count(), new_node->start(),
                        old_node->start(), old_node, alloc);
   new_node->set_finish(old_node->finish());
   old_node->set_finish(old_node->start());
   new_node->set_generation(old_node->generation());
   node_type::clear_and_delete(old_node, alloc);
 };
 const auto replace_leaf_root_node = [&](field_type new_node_size) {
   assert(iter.node_ == root());
   node_type *old_root = iter.node_;
   node_type *new_root = iter.node_ = new_leaf_root_node(new_node_size);
   transfer_and_delete(old_root, new_root);
   mutable_root() = mutable_rightmost() = new_root;
 };

 bool replaced_node = false;
 if (iter.node_->count() == max_count) {
   // Make room in the leaf for the new item.
   if (max_count < kNodeSlots) {
     // Insertion into the root where the root is smaller than the full node
     // size. Simply grow the size of the root node.
     replace_leaf_root_node(static_cast<field_type>(
         (std::min)(static_cast<int>(kNodeSlots), 2 * max_count)));
     replaced_node = true;
   } else {
     rebalance_or_split(&iter);
   }
 }
 (void)replaced_node;
#if defined(ABSL_HAVE_ADDRESS_SANITIZER) || \
   defined(ABSL_HAVE_HWADDRESS_SANITIZER)
 if (!replaced_node) {
   assert(iter.node_->is_leaf());
   if (iter.node_->is_root()) {
     replace_leaf_root_node(max_count);
   } else {
     node_type *old_node = iter.node_;
     const bool was_rightmost = rightmost() == old_node;
     const bool was_leftmost = leftmost() == old_node;
     node_type *parent = old_node->parent();
     const field_type position = old_node->position();
     node_type *new_node = iter.node_ = new_leaf_node(position, parent);
     parent->set_child_noupdate_position(position, new_node);
     transfer_and_delete(old_node, new_node);
     if (was_rightmost) mutable_rightmost() = new_node;
     // The leftmost node is stored as the parent of the root node.
     if (was_leftmost) root()->set_parent(new_node);
   }
 }
#endif
 iter.node_->emplace_value(static_cast<field_type>(iter.position_), alloc,
                           std::forward<Args>(args)...);
 assert(
     iter.node_->is_ordered_correctly(static_cast<field_type>(iter.position_),
                                      original_key_compare(key_comp())) &&
     "If this assert fails, then either (1) the comparator may violate "
     "transitivity, i.e. comp(a,b) && comp(b,c) -> comp(a,c) (see "
     "https://en.cppreference.com/w/cpp/named_req/Compare), or (2) a "
     "key may have been mutated after it was inserted into the tree.");
 ++size_;
 iter.update_generation();
 return iter;
}

template <typename P>
template <typename K>
inline auto btree<P>::internal_locate(const K &key) const
   -> SearchResult<iterator, is_key_compare_to::value> {
 iterator iter(const_cast<node_type *>(root()));
 for (;;) {
   SearchResult<size_type, is_key_compare_to::value> res =
       iter.node_->lower_bound(key, key_comp());
   iter.position_ = static_cast<int>(res.value);
   if (res.IsEq()) {
     return {iter, MatchKind::kEq};
   }
   // Note: in the non-key-compare-to case, we don't need to walk all the way
   // down the tree if the keys are equal, but determining equality would
   // require doing an extra comparison on each node on the way down, and we
   // will need to go all the way to the leaf node in the expected case.
   if (iter.node_->is_leaf()) {
     break;
   }
   iter.node_ = iter.node_->child(static_cast<field_type>(iter.position_));
 }
 // Note: in the non-key-compare-to case, the key may actually be equivalent
 // here (and the MatchKind::kNe is ignored).
 return {iter, MatchKind::kNe};
}

template <typename P>
template <typename K>
auto btree<P>::internal_lower_bound(const K &key) const
   -> SearchResult<iterator, is_key_compare_to::value> {
 if (!params_type::template can_have_multiple_equivalent_keys<K>()) {
   SearchResult<iterator, is_key_compare_to::value> ret = internal_locate(key);
   ret.value = internal_last(ret.value);
   return ret;
 }
 iterator iter(const_cast<node_type *>(root()));
 SearchResult<size_type, is_key_compare_to::value> res;
 bool seen_eq = false;
 for (;;) {
   res = iter.node_->lower_bound(key, key_comp());
   iter.position_ = static_cast<int>(res.value);
   if (iter.node_->is_leaf()) {
     break;
   }
   seen_eq = seen_eq || res.IsEq();
   iter.node_ = iter.node_->child(static_cast<field_type>(iter.position_));
 }
 if (res.IsEq()) return {iter, MatchKind::kEq};
 return {internal_last(iter), seen_eq ? MatchKind::kEq : MatchKind::kNe};
}

template <typename P>
template <typename K>
auto btree<P>::internal_upper_bound(const K &key) const -> iterator {
 iterator iter(const_cast<node_type *>(root()));
 for (;;) {
   iter.position_ = static_cast<int>(iter.node_->upper_bound(key, key_comp()));
   if (iter.node_->is_leaf()) {
     break;
   }
   iter.node_ = iter.node_->child(static_cast<field_type>(iter.position_));
 }
 return internal_last(iter);
}

template <typename P>
template <typename K>
auto btree<P>::internal_find(const K &key) const -> iterator {
 SearchResult<iterator, is_key_compare_to::value> res = internal_locate(key);
 if (res.HasMatch()) {
   if (res.IsEq()) {
     return res.value;
   }
 } else {
   const iterator iter = internal_last(res.value);
   if (iter.node_ != nullptr && !compare_keys(key, iter.key())) {
     return iter;
   }
 }
 return {nullptr, 0};
}

template <typename P>
typename btree<P>::size_type btree<P>::internal_verify(
   const node_type *node, const key_type *lo, const key_type *hi) const {
 assert(node->count() > 0);
 assert(node->count() <= node->max_count());
 if (lo) {
   assert(!compare_keys(node->key(node->start()), *lo));
 }
 if (hi) {
   assert(!compare_keys(*hi, node->key(node->finish() - 1)));
 }
 for (int i = node->start() + 1; i < node->finish(); ++i) {
   assert(!compare_keys(node->key(i), node->key(i - 1)));
 }
 size_type count = node->count();
 if (node->is_internal()) {
   for (field_type i = node->start(); i <= node->finish(); ++i) {
     assert(node->child(i) != nullptr);
     assert(node->child(i)->parent() == node);
     assert(node->child(i)->position() == i);
     count += internal_verify(node->child(i),
                              i == node->start() ? lo : &node->key(i - 1),
                              i == node->finish() ? hi : &node->key(i));
   }
 }
 return count;
}

struct btree_access {
 template <typename BtreeContainer, typename Pred>
 static auto erase_if(BtreeContainer &container, Pred pred) ->
     typename BtreeContainer::size_type {
   const auto initial_size = container.size();
   auto &tree = container.tree_;
   auto *alloc = tree.mutable_allocator();
   for (auto it = container.begin(); it != container.end();) {
     if (!pred(*it)) {
       ++it;
       continue;
     }
     auto *node = it.node_;
     if (node->is_internal()) {
       // Handle internal nodes normally.
       it = container.erase(it);
       continue;
     }
     // If this is a leaf node, then we do all the erases from this node
     // at once before doing rebalancing.

     // The current position to transfer slots to.
     int to_pos = it.position_;
     node->value_destroy(it.position_, alloc);
     while (++it.position_ < node->finish()) {
       it.update_generation();
       if (pred(*it)) {
         node->value_destroy(it.position_, alloc);
       } else {
         node->transfer(node->slot(to_pos++), node->slot(it.position_), alloc);
       }
     }
     const int num_deleted = node->finish() - to_pos;
     tree.size_ -= num_deleted;
     node->set_finish(to_pos);
     it.position_ = to_pos;
     it = tree.rebalance_after_delete(it);
   }
   return initial_size - container.size();
 }
};

#undef ABSL_BTREE_ENABLE_GENERATIONS

}  // namespace container_internal
ABSL_NAMESPACE_END
}  // namespace absl

#endif  // ABSL_CONTAINER_INTERNAL_BTREE_H_