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# AVL 树

AVL 树，是一种平衡的二叉搜索树．由于各种算法教材上对 AVL 的介绍十分冗长，造成了很多人对 AVL 树复杂、不实用的印象．但实际上，AVL 树的原理简单，实现也并不复杂．

## 性质

1. 空二叉树是一个 AVL 树
2. 如果 T 是一棵 AVL 树，那么其左右子树也是 AVL 树，并且 |ℎ(𝑙 𝑠) −ℎ(𝑟 𝑠)| ≤1|h(l s)−h(r s)|≤1，h 是其左右子树的高度
3. 树高为 𝑂(l o g⁡𝑛)O(log⁡n)

平衡因子：右子树高度 - 左子树高度

树高的证明

设 𝑓 𝑛 f n 为高度为 𝑛 n 的 AVL 树所包含的最少节点数，则有

𝑓 𝑛=⎧{ {⎨{ {⎩1(𝑛=1)2(𝑛=2)𝑓 𝑛−1+𝑓 𝑛−2+1(𝑛>2)f n={1(n=1)2(n=2)f n−1+f n−2+1(n>2)

根据常系数非齐次线性差分方程的解法，{𝑓 𝑛 +1}{f n+1} 是一个斐波那契数列．这里 𝑓 𝑛 f n 的通项为：

𝑓 𝑛=5+2√5 5(1+√5 2)𝑛+5−2√5 5(1−√5 2)𝑛−1 f n=5+2 5 5(1+5 2)n+5−2 5 5(1−5 2)n−1

斐波那契数列以指数的速度增长，对于树高 𝑛 n 有：

𝑛<l o g 1+√5 2⁡(𝑓 𝑛+1)<3 2 l o g 2⁡(𝑓 𝑛+1)n<log 1+5 2⁡(f n+1)<3 2 log 2⁡(f n+1)

因此 AVL 树的高度为 𝑂(l o g⁡𝑓 𝑛)O(log⁡f n)，这里的 𝑓 𝑛 f n 为结点数．

## 过程

### 插入结点

与 BST（二叉搜索树）中类似，先进行一次失败的查找来确定插入的位置，插入节点后根据平衡因子来决定是否需要调整．

### 删除结点

删除和 BST 类似，将结点与后继交换后再删除．

删除会导致树高以及平衡因子变化，这时需要沿着被删除结点到根的路径来调整这种变化．

### 平衡的维护

插入或删除节点后，可能会造成 AVL 树的性质 2 被破坏．因此，需要沿着从被插入/删除的节点到根的路径对树进行维护．如果对于某一个节点，性质 2 不再满足，由于我们只插入/删除了一个节点，对树高的影响不超过 1，因此该节点的平衡因子的绝对值至多为 2．由于对称性，我们在此只讨论左子树的高度比右子树大 2 的情况，即下图中 ℎ(𝐵) −ℎ(𝐸) =2 h(B)−h(E)=2．此时，还需要根据 ℎ(𝐴)h(A) 和 ℎ(𝐶)h(C) 的大小关系分两种情况讨论．需要注意的是，由于我们是自底向上维护平衡的，因此对节点 D 的所有后代来说，性质 2 仍然是被满足的．

#### 情况一：A 点树高不小于 C 点树高

设 ℎ(𝐸) =𝑥 h(E)=x，则有

⎧{ {⎨{ {⎩ℎ(𝐵)=𝑥+2 ℎ(𝐴)=𝑥+1 𝑥≤ℎ(𝐶)≤𝑥+1{h(B)=x+2 h(A)=x+1 x≤h(C)≤x+1

其中 ℎ(𝐶) ≥𝑥 h(C)≥x 是由于节点 B 满足性质 2，因此 ℎ(𝐶)h(C) 和 ℎ(𝐴)h(A) 的差不会超过 1．此时我们对节点 D 进行一次右旋操作（旋转操作与其它类型的平衡二叉搜索树相同），如下图所示．

显然节点 A、C、E 的高度不发生变化，并且有

⎧{ {⎨{ {⎩0≤ℎ(𝐶)−ℎ(𝐸)≤1 𝑥+1≤ℎ′(𝐷)=m a x(ℎ(𝐶),ℎ(𝐸))+1=ℎ(𝐶)+1≤𝑥+2 0≤ℎ′(𝐷)−ℎ(𝐴)≤1{0≤h(C)−h(E)≤1 x+1≤h′(D)=max(h(C),h(E))+1=h(C)+1≤x+2 0≤h′(D)−h(A)≤1

因此旋转后的节点 B 和 D 也满足性质 2．

#### 情况二：A 点树高小于 C 点树高

设 ℎ(𝐸) =𝑥 h(E)=x，则与刚才同理，有

⎧{ {⎨{ {⎩ℎ(𝐵)=𝑥+2 ℎ(𝐶)=𝑥+1 ℎ(𝐴)=𝑥{h(B)=x+2 h(C)=x+1 h(A)=x

此时我们先对节点 B 进行一次左旋操作，再对节点 D 进行一次右旋操作，如下图所示．

显然节点 A、E 的高度不发生变化，并且 B 的新右儿子和 D 的新左儿子分别为 C 原来的左右儿子，则有

⎧{ { { {⎨{ { { {⎩𝑥−1≤ℎ′(𝑟 𝑠 𝐵),ℎ′(𝑙 𝑠 𝐷)≤𝑥 0≤ℎ(𝐴)−ℎ′(𝑟 𝑠 𝐵)≤1 0≤ℎ(𝐸)−ℎ′(𝑙 𝑠 𝐷)≤1 ℎ′(𝐵)=m a x(ℎ(𝐴),ℎ′(𝑟 𝑠 𝐵))+1=𝑥+1 ℎ′(𝐷)=m a x(ℎ(𝐸),ℎ′(𝑙 𝑠 𝐷))+1=𝑥+1 ℎ′(𝐵)−ℎ′(𝐷)=0{x−1≤h′(r s B),h′(l s D)≤x 0≤h(A)−h′(r s B)≤1 0≤h(E)−h′(l s D)≤1 h′(B)=max(h(A),h′(r s B))+1=x+1 h′(D)=max(h(E),h′(l s D))+1=x+1 h′(B)−h′(D)=0

因此旋转后的节点 B、C、D 也满足性质 2．

维护平衡操作：伪代码
1 𝐟 𝐮 𝐧 𝐜 𝐭 𝐢 𝐨 𝐧 M a i n t a i n B a l a n c e(𝑝)2 𝑙←𝑙 𝑠 𝑝,𝑟←𝑟 𝑠 𝑝 3 𝐢 𝐟 ℎ(𝑙)−ℎ(𝑟)=2 4 𝐢 𝐟 ℎ(𝑙 𝑠 𝑙)≥ℎ(𝑟 𝑠 𝑙)5 R i g h t R o t a t e(𝑝)6 𝐞 𝐥 𝐬 𝐞 7 L e f t R o t a t e(𝑙)8 R i g h t R o t a t e(𝑝)9 𝐞 𝐥 𝐬 𝐞 𝐢 𝐟 ℎ(𝑙)−ℎ(𝑟)=−2 1 0 𝐢 𝐟 ℎ(𝑙 𝑠 𝑟)≤ℎ(𝑟 𝑠 𝑟)1 1 L e f t R o t a t e(𝑝)1 2 𝐞 𝐥 𝐬 𝐞 1 3 R i g h t R o t a t e(𝑟)1 4 L e f t R o t a t e(𝑝)1 function MaintainBalance(p)2 l←l s p,r←r s p 3 if h(l)−h(r)=2 4 if h(l s l)≥h(r s l)5 RightRotate(p)6 else 7 LeftRotate(l)8 RightRotate(p)9 else if h(l)−h(r)=−2 10 if h(l s r)≤h(r s r)11 LeftRotate(p)12 else 13 RightRotate(r)14 LeftRotate(p)

与其他平衡二叉搜索树相同，AVL 树中节点的高度、子树大小等信息需要在旋转时进行维护．

## 其他操作

AVL 树的其他操作（Predecessor、Successor、Select、Rank 等）与普通的二叉搜索树相同．

## 参考代码

下面的代码是用 AVL 树实现的 `Map`，即有序不可重映射：

参考代码

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    /**
     * @brief An AVLTree-based map implementation
     * @details The map is sorted according to the natural ordering of its
     *  keys or by a {@code Compare} function provided; This implementation
     *  provides guaranteed log(n) time cost for the contains, get, insert
     *  and remove operations.
     */

    #ifndef AVLTREE_MAP_HPP
    #define AVLTREE_MAP_HPP

    #include <cassert>
    #include <cstddef>
    #include <cstdint>
    #include <functional>
    #include <memory>
    #include <stack>
    #include <utility>
    #include <vector>

    /**
     * An AVLTree-based map implementation
     * https://en.wikipedia.org/wiki/AVL_tree
     * @tparam Key the type of keys maintained by this map
     * @tparam Value the type of mapped values
     * @tparam Compare
     */
    template <typename Key, typename Value, typename Compare = std::less<Key> >
    class AvlTreeMap {
     private:
      using USize = size_t;
      using Factor = int64_t;

      Compare compare = Compare();

     public:
      struct Entry {
        Key key;
        Value value;

        bool operator==(const Entry &rhs) const noexcept {
          return this->key == rhs.key && this->value == rhs.value;
        }

        bool operator!=(const Entry &rhs) const noexcept {
          return this->key != rhs.key || this->value != rhs.value;
        }
      };

     private:
      struct Node {
        using Ptr = std::shared_ptr<Node>;
        using Provider = const std::function<Ptr(void)> &;
        using Consumer = const std::function<void(const Ptr &)> &;

        Key key;
        Value value{};

        Ptr left = nullptr;
        Ptr right = nullptr;

        USize height = 1;

        explicit Node(Key k) : key(std::move(k)) {}

        explicit Node(Key k, Value v) : key(std::move(k)), value(std::move(v)) {}

        ~Node() = default;

        inline bool isLeaf() const noexcept {
          return this->left == nullptr && this->right == nullptr;
        }

        inline void updateHeight() noexcept {
          if (this->isLeaf()) {
            this->height = 1;
          } else if (this->left == nullptr) {
            this->height = this->right->height + 1;
          } else if (this->right == nullptr) {
            this->height = this->left->height + 1;
          } else {
            this->height = std::max(left->height, right->height) + 1;
          }
        }

        inline Factor factor() const noexcept {
          if (this->isLeaf()) {
            return 0;
          } else if (this->left == nullptr) {
            return (Factor)this->right->height;
          } else if (this->right == nullptr) {
            return (Factor) - this->left->height;
          } else {
            return (Factor)(this->right->height - this->left->height);
          }
        }

        inline Entry entry() const { return Entry{key, value}; }

        static Ptr from(const Key &k) { return std::make_shared<Node>(Node(k)); }

        static Ptr from(const Key &k, const Value &v) {
          return std::make_shared<Node>(Node(k, v));
        }
      };

      using NodePtr = typename Node::Ptr;
      using ConstNodePtr = const NodePtr &;
      using NodeProvider = typename Node::Provider;
      using NodeConsumer = typename Node::Consumer;

      NodePtr root = nullptr;
      USize count = 0;

      using K = const Key &;
      using V = const Value &;

     public:
      using EntryList = std::vector<Entry>;
      using KeyValueConsumer = const std::function<void(K, V)> &;
      using MutKeyValueConsumer = const std::function<void(K, Value &)> &;
      using KeyValueFilter = const std::function<bool(K, V)> &;

      class NoSuchMappingException : protected std::exception {
       private:
        const char *message;

       public:
        explicit NoSuchMappingException(const char *msg) : message(msg) {}

        const char *what() const noexcept override { return message; }
      };

      AvlTreeMap() noexcept = default;

      /**
       * Returns the number of entries in this map.
       * @return size_t
       */
      inline USize size() const noexcept { return this->count; }

      /**
       * Returns true if this collection contains no elements.
       * @return bool
       */
      inline bool empty() const noexcept { return this->count == 0; }

      /**
       * Removes all of the elements from this map.
       */
      void clear() noexcept {
        this->root = nullptr;
        this->count = 0;
      }

      /**
       * Returns the value to which the specified key is mapped; If this map
       * contains no mapping for the key, a {@code NoSuchMappingException} will
       * be thrown.
       * @param key
       * @return AvlTreeMap<Key, Value>::Value
       * @throws NoSuchMappingException
       */
      Value get(K key) const {
        if (this->root == nullptr) {
          throw NoSuchMappingException("Invalid key");
        } else {
          NodePtr node = this->getNode(this->root, key);
          if (node != nullptr) {
            return node->value;
          } else {
            throw NoSuchMappingException("Invalid key");
          }
        }
      }

      /**
       * Returns the value to which the specified key is mapped; If this map
       * contains no mapping for the key, a new mapping with a default value
       * will be inserted.
       * @param key
       * @return AvlTreeMap<Key, Value>::Value &
       */
      Value &getOrDefault(K key) {
        if (this->root == nullptr) {
          this->root = Node::from(key);
          this->count += 1;
          return this->root->value;
        } else {
          return this
              ->getNodeOrProvide(this->root, key,
                                 [&key]() { return Node::from(key); })
              ->value;
        }
      }

      /**
       * Returns true if this map contains a mapping for the specified key.
       * @param key
       * @return bool
       */
      bool contains(K key) const {
        return this->getNode(this->root, key) != nullptr;
      }

      /**
       * Associates the specified value with the specified key in this map.
       * @param key
       * @param value
       */
      void insert(K key, V value) {
        if (this->root == nullptr) {
          this->root = Node::from(key, value);
          this->count += 1;
        } else {
          this->insert(this->root, key, value);
        }
      }

      /**
       * If the specified key is not already associated with a value, associates
       * it with the given value and returns true, else returns false.
       * @param key
       * @param value
       * @return bool
       */
      bool insertIfAbsent(K key, V value) {
        USize sizeBeforeInsertion = this->size();
        if (this->root == nullptr) {
          this->root = Node::from(key, value);
          this->count += 1;
        } else {
          this->insert(this->root, key, value, false);
        }
        return this->size() > sizeBeforeInsertion;
      }

      /**
       * If the specified key is not already associated with a value, associates
       * it with the given value and returns the value, else returns the associated
       * value.
       * @param key
       * @param value
       * @return
       */
      Value &getOrInsert(K key, V value) {
        if (this->root == nullptr) {
          this->root = Node::from(key, value);
          this->count += 1;
          return root->value;
        } else {
          NodePtr node = getNodeOrProvide(this->root, key,
                                          [&]() { return Node::from(key, value); });
          return node->value;
        }
      }

      Value operator[](K key) const { return this->get(key); }

      Value &operator[](K key) { return this->getOrDefault(key); }

      /**
       * Removes the mapping for a key from this map if it is present;
       * Returns true if the mapping is present else returns false
       * @param key the key of the mapping
       * @return bool
       */
      bool remove(K key) {
        if (this->root == nullptr) {
          return false;
        } else {
          return this->remove(this->root, key, [](ConstNodePtr) {});
        }
      }

      /**
       * Removes the mapping for a key from this map if it is present and returns
       * the value which is mapped to the key; If this map contains no mapping for
       * the key, a {@code NoSuchMappingException} will be thrown.
       * @param key
       * @return AvlTreeMap<Key, Value>::Value
       * @throws NoSuchMappingException
       */
      Value getAndRemove(K key) {
        Value result;
        NodeConsumer action = [&](ConstNodePtr node) { result = node->value; };

        if (root == nullptr) {
          throw NoSuchMappingException("Invalid key");
        } else {
          if (remove(this->root, key, action)) {
            return result;
          } else {
            throw NoSuchMappingException("Invalid key");
          }
        }
      }

      /**
       * Gets the entry corresponding to the specified key; if no such entry
       * exists, returns the entry for the least key greater than the specified
       * key; if no such entry exists (i.e., the greatest key in the Tree is less
       * than the specified key), a {@code NoSuchMappingException} will be thrown.
       * @param key
       * @return AvlTreeMap<Key, Value>::Entry
       * @throws NoSuchMappingException
       */
      Entry getCeilingEntry(K key) const {
        if (this->root == nullptr) {
          throw NoSuchMappingException("No ceiling entry in this map");
        }

        NodePtr node = this->root;
        std::stack<NodePtr> ancestors;

        while (node != nullptr) {
          if (key == node->key) {
            return node->entry();
          }

          if (compare(key, node->key)) {
            /* key < node->key */
            if (node->left != nullptr) {
              ancestors.push(node);
              node = node->left;
            } else {
              return node->entry();
            }
          } else {
            /* key > node->key */
            if (node->right != nullptr) {
              ancestors.push(node);
              node = node->right;
            } else {
              if (ancestors.empty()) {
                throw NoSuchMappingException("No ceiling entry in this map");
              }

              NodePtr parent = ancestors.top();
              ancestors.pop();

              while (node == parent->right) {
                node = parent;
                if (!ancestors.empty()) {
                  parent = ancestors.top();
                  ancestors.pop();
                } else {
                  throw NoSuchMappingException("No ceiling entry in this map");
                }
              }

              return parent->entry();
            }
          }
        }

        throw NoSuchMappingException("No ceiling entry in this map");
      }

      /**
       * Gets the entry corresponding to the specified key; if no such entry exists,
       * returns the entry for the greatest key less than the specified key;
       * if no such entry exists, a {@code NoSuchMappingException} will be thrown.
       * @param key
       * @return AvlTreeMap<Key, Value>::Entry
       * @throws NoSuchMappingException
       */
      Entry getFloorEntry(K key) const {
        if (this->root == nullptr) {
          throw NoSuchMappingException("No floor entry exists in this map");
        }

        NodePtr node = this->root;
        std::stack<NodePtr> ancestors;

        while (node != nullptr) {
          if (key == node->key) {
            return node->entry();
          }

          if (compare(key, node->key)) {
            /* key < node->key */
            if (node->left != nullptr) {
              ancestors.push(node);
              node = node->left;
            } else {
              if (ancestors.empty()) {
                throw NoSuchMappingException("No floor entry exists in this map");
              }

              NodePtr parent = ancestors.top();
              ancestors.pop();

              while (node == parent->left) {
                node = parent;
                if (!ancestors.empty()) {
                  parent = ancestors.top();
                  ancestors.pop();
                } else {
                  throw NoSuchMappingException("No floor entry exists in this map");
                }
              }

              return parent->entry();
            }
          } else {
            /* key > node->key */
            if (node->right != nullptr) {
              ancestors.push(node);
              node = node->right;
            } else {
              return node->entry();
            }
          }
        }

        throw NoSuchMappingException("No floor entry exists in this map");
      }

      /**
       * Gets the entry for the least key greater than the specified
       * key; if no such entry exists, returns the entry for the least
       * key greater than the specified key; if no such entry exists,
       * a {@code NoSuchMappingException} will be thrown.
       * @param key
       * @return AvlTreeMap<Key, Value>::Entry
       * @throws NoSuchMappingException
       */
      Entry getHigherEntry(K key) {
        if (this->root == nullptr) {
          throw NoSuchMappingException("No higher entry exists in this map");
        }

        NodePtr node = this->root;
        std::stack<NodePtr> ancestors;

        while (node != nullptr) {
          if (compare(key, node->key)) {
            /* key < node->key */
            if (node->left != nullptr) {
              ancestors.push(node);
              node = node->left;
            } else {
              return node->entry();
            }
          } else {
            /* key >= node->key */
            if (node->right != nullptr) {
              ancestors.push(node);
              node = node->right;
            } else {
              if (ancestors.empty()) {
                throw NoSuchMappingException("No higher entry exists in this map");
              }

              NodePtr parent = ancestors.top();
              ancestors.pop();

              while (node == parent->right) {
                node = parent;
                if (!ancestors.empty()) {
                  parent = ancestors.top();
                  ancestors.pop();
                } else {
                  throw NoSuchMappingException(
                      "No higher entry exists in this map");
                }
              }

              return parent->entry();
            }
          }
        }

        throw NoSuchMappingException("No higher entry exists in this map");
      }

      /**
       * Returns the entry for the greatest key less than the specified key; if
       * no such entry exists (i.e., the least key in the Tree is greater than
       * the specified key), a {@code NoSuchMappingException} will be thrown.
       * @param key
       * @return AvlTreeMap<Key, Value>::Entry
       * @throws NoSuchMappingException
       */
      Entry getLowerEntry(K key) const {
        if (this->root == nullptr) {
          throw NoSuchMappingException("No lower entry exists in this map");
        }

        NodePtr node = this->root;
        std::stack<NodePtr> ancestors;

        while (node != nullptr) {
          if (compare(key, node->key) || key == node->key) {
            /* key <= node->key */
            if (node->left != nullptr) {
              ancestors.push(node);
              node = node->left;
            } else {
              if (ancestors.empty()) {
                throw NoSuchMappingException("No lower entry exists in this map");
              }

              NodePtr parent = ancestors.top();
              ancestors.pop();

              while (node == parent->left) {
                node = parent;
                if (!ancestors.empty()) {
                  parent = ancestors.top();
                  ancestors.pop();
                } else {
                  throw NoSuchMappingException("No lower entry exists in this map");
                }
              }

              return parent->entry();
            }
          } else {
            /* key > node->key */
            if (node->right != nullptr) {
              ancestors.push(node);
              node = node->right;
            } else {
              return node->entry();
            }
          }
        }

        throw NoSuchMappingException("No lower entry exists in this map");
      }

      /**
       * Remove all entries that satisfy the filter condition.
       * @param filter
       */
      void removeAll(KeyValueFilter filter) {
        std::vector<Key> keys;
        this->inorderTraversal([&](ConstNodePtr node) {
          if (filter(node->key, node->value)) {
            keys.push_back(node->key);
          }
        });
        for (const Key &key : keys) {
          this->remove(key);
        }
      }

      /**
       * Performs the given action for each key and value entry in this map.
       * The value is immutable for the action.
       * @param action
       */
      void forEach(KeyValueConsumer action) const {
        this->inorderTraversal(
            [&](ConstNodePtr node) { action(node->key, node->value); });
      }

      /**
       * Performs the given action for each key and value entry in this map.
       * The value is mutable for the action.
       * @param action
       */
      void forEachMut(MutKeyValueConsumer action) {
        this->inorderTraversal(
            [&](ConstNodePtr node) { action(node->key, node->value); });
      }

      /**
       * Returns a list containing all of the entries in this map.
       * @return AvlTreeMap<Key, Value>::EntryList
       */
      EntryList toEntryList() const {
        EntryList entryList;
        this->inorderTraversal(
            [&](ConstNodePtr node) { entryList.push_back(node->entry()); });
        return entryList;
      }

     private:
      static NodePtr rotateLeft(ConstNodePtr node) {
        // clang-format off
        //     |                       |
        //     N                       S
        //    / \     l-rotate(N)     / \
        //   L   S    ==========>    N   R
        //      / \                 / \
        //     M   R               L   M
        NodePtr successor = node->right;
        // clang-format on
        node->right = successor->left;
        successor->left = node;

        node->updateHeight();
        successor->updateHeight();

        return successor;
      }

      static NodePtr rotateRight(ConstNodePtr node) {
        // clang-format off
        //       |                   |
        //       N                   S
        //      / \   r-rotate(N)   / \
        //     S   R  ==========>  L   N
        //    / \                     / \
        //   L   M                   M   R
        NodePtr successor = node->left;
        // clang-format on
        node->left = successor->right;
        successor->right = node;

        node->updateHeight();
        successor->updateHeight();

        return successor;
      }

      static void swapNode(NodePtr &lhs, NodePtr &rhs) {
        std::swap(lhs->key, rhs->key);
        std::swap(lhs->value, rhs->value);
        std::swap(lhs, rhs);
      }

      static void fixBalance(NodePtr &node) {
        if (node->factor() < -1) {
          if (node->left->factor() < 0) {
            // clang-format off
            //  Left-Left Case
            //       |
            //       C                 |
            //      /   r-rotate(C)    B
            //     B    ==========>   / \
            //    /                  A   C
            //   A
            // clang-format on
            node = rotateRight(node);
          } else {
            // clang-format off
            //  Left-Right Case
            //     |                   |
            //     C                   C                 |
            //    /   l-rotate(A)     /   r-rotate(C)    B
            //   A    ==========>    B    ==========>   / \
            //    \                 /                  A   C
            //     B               A
            // clang-format on
            node->left = rotateLeft(node->left);
            node = rotateRight(node);
          }
        } else if (node->factor() > 1) {
          if (node->right->factor() > 0) {
            // clang-format off
            //  Right-Right Case
            //   |
            //   C                     |
            //    \     l-rotate(C)    B
            //     B    ==========>   / \
            //      \                A   C
            //       A
            // clang-format on
            node = rotateLeft(node);
          } else {
            // clang-format off
            //  Right-Left Case
            //   |                 |
            //   A                 A                     |
            //    \   r-rotate(C)   \     l-rotate(A)    B
            //     C  ==========>    B    ==========>   / \
            //    /                   \                A   C
            //   B                     C
            // clang-format on
            node->right = rotateRight(node->right);
            node = rotateLeft(node);
          }
        }
      }

      NodePtr getNodeOrProvide(NodePtr &node, K key, NodeProvider provide) {
        assert(node != nullptr);

        if (key == node->key) {
          return node;
        }

        assert(key != node->key);

        NodePtr result;

        if (compare(key, node->key)) {
          /* key < node->key */
          if (node->left == nullptr) {
            result = node->left = provide();
            this->count += 1;
            node->updateHeight();
          } else {
            result = getNodeOrProvide(node->left, key, provide);
            node->updateHeight();
            fixBalance(node);
          }
        } else {
          /* key > node->key */
          if (node->right == nullptr) {
            result = node->right = provide();
            this->count += 1;
            node->updateHeight();
          } else {
            result = getNodeOrProvide(node->right, key, provide);
            node->updateHeight();
            fixBalance(node);
          }
        }

        return result;
      }

      NodePtr getNode(ConstNodePtr node, K key) const {
        assert(node != nullptr);

        if (key == node->key) {
          return node;
        }

        if (compare(key, node->key)) {
          /* key < node->key */
          return node->left == nullptr ? nullptr : getNode(node->left, key);
        } else {
          /* key > node->key */
          return node->right == nullptr ? nullptr : getNode(node->right, key);
        }
      }

      void insert(NodePtr &node, K key, V value, bool replace = true) {
        assert(node != nullptr);

        if (key == node->key) {
          if (replace) {
            node->value = value;
          }
          return;
        }

        assert(key != node->key);

        if (compare(key, node->key)) {
          /* key < node->key */
          if (node->left == nullptr) {
            node->left = Node::from(key, value);
            this->count += 1;
            node->updateHeight();
          } else {
            insert(node->left, key, value, replace);
            node->updateHeight();
            fixBalance(node);
          }
        } else {
          /* key > node->key */
          if (node->right == nullptr) {
            node->right = Node::from(key, value);
            this->count += 1;
            node->updateHeight();
          } else {
            insert(node->right, key, value, replace);
            node->updateHeight();
            fixBalance(node);
          }
        }
      }

      bool remove(NodePtr &node, K key, NodeConsumer action) {
        assert(node != nullptr);

        if (key != node->key) {
          if (compare(key, node->key)) {
            /* key < node->key */
            NodePtr &left = node->left;
            if (left != nullptr && remove(left, key, action)) {
              node->updateHeight();
              fixBalance(node);
              return true;
            } else {
              return false;
            }
          } else {
            /* key > node->key */
            NodePtr &right = node->right;
            if (right != nullptr && remove(right, key, action)) {
              node->updateHeight();
              fixBalance(node);
              return true;
            } else {
              return false;
            }
          }
        }

        assert(key == node->key);
        action(node);

        if (node->isLeaf()) {
          // Case 1: no child
          node = nullptr;
        } else if (node->right == nullptr) {
          // clang-format off
          // Case 2: left child only
          //     P
          //     |  remove(N)  P
          //     N  ========>  |
          //    /              L
          //   L
          // clang-format on
          node = node->left;
          node->updateHeight();
        } else if (node->left == nullptr) {
          // clang-format off
          // Case 3: right child only
          //   P
          //   |    remove(N)  P
          //   N    ========>  |
          //    \              R
          //     R
          // clang-format on
          node = node->right;
          node->updateHeight();
        } else if (node->right->left == nullptr) {
          // clang-format off
          // Case 4: both left and right child, right child has no left child
          //    |                 |
          //    N    remove(N)    R
          //   / \   ========>   /
          //  L   R             L
          // clang-format on
          NodePtr right = node->right;
          swapNode(node, right);
          right->right = node->right;
          node = right;
          node->updateHeight();
          fixBalance(node);
        } else {
          // clang-format off
          // Case 5: both left and right child, right child is not a leaf
          //   Step 1. find the node N with the smallest key
          //           and its parent P on the right subtree
          //   Step 2. swap S and N
          //   Step 3. remove node N like Case 1 or Case 3
          //   Step 4. update height for P
          //     |                  |
          //     N                  S                 |
          //    / \                / \                S
          //   L  ..  swap(N, S)  L  ..  remove(N)   / \
          //       |  =========>      |  ========>  L  ..
          //       P                  P                 |
          //      / \                / \                P
          //     S  ..              N  ..              / \
          //      \                  \                R  ..
          //       R                  R
          // clang-format on

          // Step 1
          NodePtr successor = node->right;
          NodePtr parent = node;
          while (successor->left != nullptr) {
            parent = successor;
            successor = parent->left;
          }
          // Step 2
          swapNode(node, successor);
          // Step 3
          parent->left = node->right;
          // Restore node
          node = successor;
          // Step 4
          parent->updateHeight();
        }

        this->count -= 1;
        return true;
      }

      void inorderTraversal(NodeConsumer action) const {
        if (this->root == nullptr) {
          return;
        }

        std::stack<NodePtr> stack;
        NodePtr node = this->root;

        while (node != nullptr || !stack.empty()) {
          while (node != nullptr) {
            stack.push(node);
            node = node->left;
          }
          if (!stack.empty()) {
            node = stack.top();
            stack.pop();
            action(node);
            node = node->right;
          }
        }
      }
    };

    #endif  // AVLTREE_MAP_HPP

## 其他资料

在 AVL Tree Visualization 可以观察 AVL 树维护平衡的过程．

维基百科 -- AVL 树

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