---
comments: true
---

# 7.5   AVL tree *

In the "Binary Search Tree" section, we mentioned that after multiple insertions and removals, a binary search tree might degrade to a linked list. In such cases, the time complexity of all operations degrades from $O(\log n)$ to $O(n)$.

As shown in Figure 7-24, after two node removal operations, this binary search tree will degrade into a linked list.

![Degradation of an AVL tree after removing nodes](avl_tree.assets/avltree_degradation_from_removing_node.png){ class="animation-figure" }

<p align="center"> Figure 7-24 &nbsp; Degradation of an AVL tree after removing nodes </p>

For example, in the perfect binary tree shown in Figure 7-25, after inserting two nodes, the tree will lean heavily to the left, and the time complexity of search operations will also degrade.

![Degradation of an AVL tree after inserting nodes](avl_tree.assets/avltree_degradation_from_inserting_node.png){ class="animation-figure" }

<p align="center"> Figure 7-25 &nbsp; Degradation of an AVL tree after inserting nodes </p>

In 1962, G. M. Adelson-Velsky and E. M. Landis proposed the <u>AVL Tree</u> in their paper "An algorithm for the organization of information". The paper detailed a series of operations to ensure that after continuously adding and removing nodes, the AVL tree would not degrade, thus maintaining the time complexity of various operations at $O(\log n)$ level. In other words, in scenarios where frequent additions, removals, searches, and modifications are needed, the AVL tree can always maintain efficient data operation performance, which has great application value.

## 7.5.1 &nbsp; Common terminology in AVL trees

An AVL tree is both a binary search tree and a balanced binary tree, satisfying all properties of these two types of binary trees, hence it is a <u>balanced binary search tree</u>.

### 1. &nbsp; Node height

Since the operations related to AVL trees require obtaining node heights, we need to add a `height` variable to the node class:

=== "Python"

    ```python title=""
    class TreeNode:
        """AVL tree node"""
        def __init__(self, val: int):
            self.val: int = val                 # Node value
            self.height: int = 0                # Node height
            self.left: TreeNode | None = None   # Left child reference
            self.right: TreeNode | None = None  # Right child reference
    ```

=== "C++"

    ```cpp title=""
    /* AVL tree node */
    struct TreeNode {
        int val{};          // Node value
        int height = 0;     // Node height
        TreeNode *left{};   // Left child
        TreeNode *right{};  // Right child
        TreeNode() = default;
        explicit TreeNode(int x) : val(x){}
    };
    ```

=== "Java"

    ```java title=""
    /* AVL tree node */
    class TreeNode {
        public int val;        // Node value
        public int height;     // Node height
        public TreeNode left;  // Left child
        public TreeNode right; // Right child
        public TreeNode(int x) { val = x; }
    }
    ```

=== "C#"

    ```csharp title=""
    /* AVL tree node */
    class TreeNode(int? x) {
        public int? val = x;    // Node value
        public int height;      // Node height
        public TreeNode? left;  // Left child reference
        public TreeNode? right; // Right child reference
    }
    ```

=== "Go"

    ```go title=""
    /* AVL tree node */
    type TreeNode struct {
        Val    int       // Node value
        Height int       // Node height
        Left   *TreeNode // Left child reference
        Right  *TreeNode // Right child reference
    }
    ```

=== "Swift"

    ```swift title=""
    /* AVL tree node */
    class TreeNode {
        var val: Int // Node value
        var height: Int // Node height
        var left: TreeNode? // Left child
        var right: TreeNode? // Right child

        init(x: Int) {
            val = x
            height = 0
        }
    }
    ```

=== "JS"

    ```javascript title=""
    /* AVL tree node */
    class TreeNode {
        val; // Node value
        height; // Node height
        left; // Left child pointer
        right; // Right child pointer
        constructor(val, left, right, height) {
            this.val = val === undefined ? 0 : val;
            this.height = height === undefined ? 0 : height;
            this.left = left === undefined ? null : left;
            this.right = right === undefined ? null : right;
        }
    }
    ```

=== "TS"

    ```typescript title=""
    /* AVL tree node */
    class TreeNode {
        val: number;            // Node value
        height: number;         // Node height
        left: TreeNode | null;  // Left child pointer
        right: TreeNode | null; // Right child pointer
        constructor(val?: number, height?: number, left?: TreeNode | null, right?: TreeNode | null) {
            this.val = val === undefined ? 0 : val;
            this.height = height === undefined ? 0 : height; 
            this.left = left === undefined ? null : left; 
            this.right = right === undefined ? null : right; 
        }
    }
    ```

=== "Dart"

    ```dart title=""
    /* AVL tree node */
    class TreeNode {
      int val;         // Node value
      int height;      // Node height
      TreeNode? left;  // Left child
      TreeNode? right; // Right child
      TreeNode(this.val, [this.height = 0, this.left, this.right]);
    }
    ```

=== "Rust"

    ```rust title=""
    use std::rc::Rc;
    use std::cell::RefCell;

    /* AVL tree node */
    struct TreeNode {
        val: i32,                               // Node value
        height: i32,                            // Node height
        left: Option<Rc<RefCell<TreeNode>>>,    // Left child
        right: Option<Rc<RefCell<TreeNode>>>,   // Right child
    }

    impl TreeNode {
        /* Constructor */
        fn new(val: i32) -> Rc<RefCell<Self>> {
            Rc::new(RefCell::new(Self {
                val,
                height: 0,
                left: None,
                right: None
            }))
        }
    }
    ```

=== "C"

    ```c title=""
    /* AVL tree node */
    TreeNode struct TreeNode {
        int val;
        int height;
        struct TreeNode *left;
        struct TreeNode *right;
    } TreeNode;

    /* Constructor */
    TreeNode *newTreeNode(int val) {
        TreeNode *node;

        node = (TreeNode *)malloc(sizeof(TreeNode));
        node->val = val;
        node->height = 0;
        node->left = NULL;
        node->right = NULL;
        return node;
    }
    ```

=== "Kotlin"

    ```kotlin title=""
    /* AVL tree node */
    class TreeNode(val _val: Int) {  // Node value
        val height: Int = 0          // Node height
        val left: TreeNode? = null   // Left child
        val right: TreeNode? = null  // Right child
    }
    ```

=== "Ruby"

    ```ruby title=""

    ```

=== "Zig"

    ```zig title=""

    ```

The "node height" refers to the distance from that node to its farthest leaf node, i.e., the number of "edges" passed. It is important to note that the height of a leaf node is $0$, and the height of a null node is $-1$. We will create two utility functions for getting and updating the height of a node:

=== "Python"

    ```python title="avl_tree.py"
    def height(self, node: TreeNode | None) -> int:
        """获取节点高度"""
        # 空节点高度为 -1 ，叶节点高度为 0
        if node is not None:
            return node.height
        return -1

    def update_height(self, node: TreeNode | None):
        """更新节点高度"""
        # 节点高度等于最高子树高度 + 1
        node.height = max([self.height(node.left), self.height(node.right)]) + 1
    ```

=== "C++"

    ```cpp title="avl_tree.cpp"
    /* 获取节点高度 */
    int height(TreeNode *node) {
        // 空节点高度为 -1 ，叶节点高度为 0
        return node == nullptr ? -1 : node->height;
    }

    /* 更新节点高度 */
    void updateHeight(TreeNode *node) {
        // 节点高度等于最高子树高度 + 1
        node->height = max(height(node->left), height(node->right)) + 1;
    }
    ```

=== "Java"

    ```java title="avl_tree.java"
    /* 获取节点高度 */
    int height(TreeNode node) {
        // 空节点高度为 -1 ，叶节点高度为 0
        return node == null ? -1 : node.height;
    }

    /* 更新节点高度 */
    void updateHeight(TreeNode node) {
        // 节点高度等于最高子树高度 + 1
        node.height = Math.max(height(node.left), height(node.right)) + 1;
    }
    ```

=== "C#"

    ```csharp title="avl_tree.cs"
    /* 获取节点高度 */
    int Height(TreeNode? node) {
        // 空节点高度为 -1 ，叶节点高度为 0
        return node == null ? -1 : node.height;
    }

    /* 更新节点高度 */
    void UpdateHeight(TreeNode node) {
        // 节点高度等于最高子树高度 + 1
        node.height = Math.Max(Height(node.left), Height(node.right)) + 1;
    }
    ```

=== "Go"

    ```go title="avl_tree.go"
    /* 获取节点高度 */
    func (t *aVLTree) height(node *TreeNode) int {
        // 空节点高度为 -1 ，叶节点高度为 0
        if node != nil {
            return node.Height
        }
        return -1
    }

    /* 更新节点高度 */
    func (t *aVLTree) updateHeight(node *TreeNode) {
        lh := t.height(node.Left)
        rh := t.height(node.Right)
        // 节点高度等于最高子树高度 + 1
        if lh > rh {
            node.Height = lh + 1
        } else {
            node.Height = rh + 1
        }
    }
    ```

=== "Swift"

    ```swift title="avl_tree.swift"
    /* 获取节点高度 */
    func height(node: TreeNode?) -> Int {
        // 空节点高度为 -1 ，叶节点高度为 0
        node?.height ?? -1
    }

    /* 更新节点高度 */
    func updateHeight(node: TreeNode?) {
        // 节点高度等于最高子树高度 + 1
        node?.height = max(height(node: node?.left), height(node: node?.right)) + 1
    }
    ```

=== "JS"

    ```javascript title="avl_tree.js"
    /* 获取节点高度 */
    height(node) {
        // 空节点高度为 -1 ，叶节点高度为 0
        return node === null ? -1 : node.height;
    }

    /* 更新节点高度 */
    #updateHeight(node) {
        // 节点高度等于最高子树高度 + 1
        node.height =
            Math.max(this.height(node.left), this.height(node.right)) + 1;
    }
    ```

=== "TS"

    ```typescript title="avl_tree.ts"
    /* 获取节点高度 */
    height(node: TreeNode): number {
        // 空节点高度为 -1 ，叶节点高度为 0
        return node === null ? -1 : node.height;
    }

    /* 更新节点高度 */
    updateHeight(node: TreeNode): void {
        // 节点高度等于最高子树高度 + 1
        node.height =
            Math.max(this.height(node.left), this.height(node.right)) + 1;
    }
    ```

=== "Dart"

    ```dart title="avl_tree.dart"
    /* 获取节点高度 */
    int height(TreeNode? node) {
      // 空节点高度为 -1 ，叶节点高度为 0
      return node == null ? -1 : node.height;
    }

    /* 更新节点高度 */
    void updateHeight(TreeNode? node) {
      // 节点高度等于最高子树高度 + 1
      node!.height = max(height(node.left), height(node.right)) + 1;
    }
    ```

=== "Rust"

    ```rust title="avl_tree.rs"
    /* 获取节点高度 */
    fn height(node: OptionTreeNodeRc) -> i32 {
        // 空节点高度为 -1 ，叶节点高度为 0
        match node {
            Some(node) => node.borrow().height,
            None => -1,
        }
    }

    /* 更新节点高度 */
    fn update_height(node: OptionTreeNodeRc) {
        if let Some(node) = node {
            let left = node.borrow().left.clone();
            let right = node.borrow().right.clone();
            // 节点高度等于最高子树高度 + 1
            node.borrow_mut().height = std::cmp::max(Self::height(left), Self::height(right)) + 1;
        }
    }
    ```

=== "C"

    ```c title="avl_tree.c"
    /* 获取节点高度 */
    int height(TreeNode *node) {
        // 空节点高度为 -1 ，叶节点高度为 0
        if (node != NULL) {
            return node->height;
        }
        return -1;
    }

    /* 更新节点高度 */
    void updateHeight(TreeNode *node) {
        int lh = height(node->left);
        int rh = height(node->right);
        // 节点高度等于最高子树高度 + 1
        if (lh > rh) {
            node->height = lh + 1;
        } else {
            node->height = rh + 1;
        }
    }
    ```

=== "Kotlin"

    ```kotlin title="avl_tree.kt"
    /* 获取节点高度 */
    fun height(node: TreeNode?): Int {
        // 空节点高度为 -1 ，叶节点高度为 0
        return node?.height ?: -1
    }

    /* 更新节点高度 */
    fun updateHeight(node: TreeNode?) {
        // 节点高度等于最高子树高度 + 1
        node?.height = max(height(node?.left), height(node?.right)) + 1
    }
    ```

=== "Ruby"

    ```ruby title="avl_tree.rb"
    ### 获取节点高度 ###
    def height(node)
      # 空节点高度为 -1 ，叶节点高度为 0
      return node.height unless node.nil?

      -1
    end

    ### 更新节点高度 ###
    def update_height(node)
      # 节点高度等于最高子树高度 + 1
      node.height = [height(node.left), height(node.right)].max + 1
    end
    ```

=== "Zig"

    ```zig title="avl_tree.zig"
    // 获取节点高度
    fn height(self: *Self, node: ?*inc.TreeNode(T)) i32 {
        _ = self;
        // 空节点高度为 -1 ，叶节点高度为 0
        return if (node == null) -1 else node.?.height;
    }

    // 更新节点高度
    fn updateHeight(self: *Self, node: ?*inc.TreeNode(T)) void {
        // 节点高度等于最高子树高度 + 1
        node.?.height = @max(self.height(node.?.left), self.height(node.?.right)) + 1;
    }
    ```

### 2. &nbsp; Node balance factor

The <u>balance factor</u> of a node is defined as the height of the node's left subtree minus the height of its right subtree, with the balance factor of a null node defined as $0$. We will also encapsulate the functionality of obtaining the node balance factor into a function for easy use later on:

=== "Python"

    ```python title="avl_tree.py"
    def balance_factor(self, node: TreeNode | None) -> int:
        """获取平衡因子"""
        # 空节点平衡因子为 0
        if node is None:
            return 0
        # 节点平衡因子 = 左子树高度 - 右子树高度
        return self.height(node.left) - self.height(node.right)
    ```

=== "C++"

    ```cpp title="avl_tree.cpp"
    /* 获取平衡因子 */
    int balanceFactor(TreeNode *node) {
        // 空节点平衡因子为 0
        if (node == nullptr)
            return 0;
        // 节点平衡因子 = 左子树高度 - 右子树高度
        return height(node->left) - height(node->right);
    }
    ```

=== "Java"

    ```java title="avl_tree.java"
    /* 获取平衡因子 */
    int balanceFactor(TreeNode node) {
        // 空节点平衡因子为 0
        if (node == null)
            return 0;
        // 节点平衡因子 = 左子树高度 - 右子树高度
        return height(node.left) - height(node.right);
    }
    ```

=== "C#"

    ```csharp title="avl_tree.cs"
    /* 获取平衡因子 */
    int BalanceFactor(TreeNode? node) {
        // 空节点平衡因子为 0
        if (node == null) return 0;
        // 节点平衡因子 = 左子树高度 - 右子树高度
        return Height(node.left) - Height(node.right);
    }
    ```

=== "Go"

    ```go title="avl_tree.go"
    /* 获取平衡因子 */
    func (t *aVLTree) balanceFactor(node *TreeNode) int {
        // 空节点平衡因子为 0
        if node == nil {
            return 0
        }
        // 节点平衡因子 = 左子树高度 - 右子树高度
        return t.height(node.Left) - t.height(node.Right)
    }
    ```

=== "Swift"

    ```swift title="avl_tree.swift"
    /* 获取平衡因子 */
    func balanceFactor(node: TreeNode?) -> Int {
        // 空节点平衡因子为 0
        guard let node = node else { return 0 }
        // 节点平衡因子 = 左子树高度 - 右子树高度
        return height(node: node.left) - height(node: node.right)
    }
    ```

=== "JS"

    ```javascript title="avl_tree.js"
    /* 获取平衡因子 */
    balanceFactor(node) {
        // 空节点平衡因子为 0
        if (node === null) return 0;
        // 节点平衡因子 = 左子树高度 - 右子树高度
        return this.height(node.left) - this.height(node.right);
    }
    ```

=== "TS"

    ```typescript title="avl_tree.ts"
    /* 获取平衡因子 */
    balanceFactor(node: TreeNode): number {
        // 空节点平衡因子为 0
        if (node === null) return 0;
        // 节点平衡因子 = 左子树高度 - 右子树高度
        return this.height(node.left) - this.height(node.right);
    }
    ```

=== "Dart"

    ```dart title="avl_tree.dart"
    /* 获取平衡因子 */
    int balanceFactor(TreeNode? node) {
      // 空节点平衡因子为 0
      if (node == null) return 0;
      // 节点平衡因子 = 左子树高度 - 右子树高度
      return height(node.left) - height(node.right);
    }
    ```

=== "Rust"

    ```rust title="avl_tree.rs"
    /* 获取平衡因子 */
    fn balance_factor(node: OptionTreeNodeRc) -> i32 {
        match node {
            // 空节点平衡因子为 0
            None => 0,
            // 节点平衡因子 = 左子树高度 - 右子树高度
            Some(node) => {
                Self::height(node.borrow().left.clone()) - Self::height(node.borrow().right.clone())
            }
        }
    }
    ```

=== "C"

    ```c title="avl_tree.c"
    /* 获取平衡因子 */
    int balanceFactor(TreeNode *node) {
        // 空节点平衡因子为 0
        if (node == NULL) {
            return 0;
        }
        // 节点平衡因子 = 左子树高度 - 右子树高度
        return height(node->left) - height(node->right);
    }
    ```

=== "Kotlin"

    ```kotlin title="avl_tree.kt"
    /* 获取平衡因子 */
    fun balanceFactor(node: TreeNode?): Int {
        // 空节点平衡因子为 0
        if (node == null) return 0
        // 节点平衡因子 = 左子树高度 - 右子树高度
        return height(node.left) - height(node.right)
    }
    ```

=== "Ruby"

    ```ruby title="avl_tree.rb"
    ### 获取平衡因子 ###
    def balance_factor(node)
      # 空节点平衡因子为 0
      return 0 if node.nil?

      # 节点平衡因子 = 左子树高度 - 右子树高度
      height(node.left) - height(node.right)
    end
    ```

=== "Zig"

    ```zig title="avl_tree.zig"
    // 获取平衡因子
    fn balanceFactor(self: *Self, node: ?*inc.TreeNode(T)) i32 {
        // 空节点平衡因子为 0
        if (node == null) return 0;
        // 节点平衡因子 = 左子树高度 - 右子树高度
        return self.height(node.?.left) - self.height(node.?.right);
    }
    ```

!!! tip

    Let the balance factor be $f$, then the balance factor of any node in an AVL tree satisfies $-1 \le f \le 1$.

## 7.5.2 &nbsp; Rotations in AVL trees

The characteristic feature of an AVL tree is the "rotation" operation, which can restore balance to an unbalanced node without affecting the in-order traversal sequence of the binary tree. In other words, **the rotation operation can maintain the property of a "binary search tree" while also turning the tree back into a "balanced binary tree"**.

We call nodes with an absolute balance factor $> 1$ "unbalanced nodes". Depending on the type of imbalance, there are four kinds of rotations: right rotation, left rotation, right-left rotation, and left-right rotation. Below, we detail these rotation operations.

### 1. &nbsp; Right rotation

As shown in Figure 7-26, the first unbalanced node from the bottom up in the binary tree is "node 3". Focusing on the subtree with this unbalanced node as the root, denoted as `node`, and its left child as `child`, perform a "right rotation". After the right rotation, the subtree is balanced again while still maintaining the properties of a binary search tree.

=== "<1>"
    ![Steps of right rotation](avl_tree.assets/avltree_right_rotate_step1.png){ class="animation-figure" }

=== "<2>"
    ![avltree_right_rotate_step2](avl_tree.assets/avltree_right_rotate_step2.png){ class="animation-figure" }

=== "<3>"
    ![avltree_right_rotate_step3](avl_tree.assets/avltree_right_rotate_step3.png){ class="animation-figure" }

=== "<4>"
    ![avltree_right_rotate_step4](avl_tree.assets/avltree_right_rotate_step4.png){ class="animation-figure" }

<p align="center"> Figure 7-26 &nbsp; Steps of right rotation </p>

As shown in Figure 7-27, when the `child` node has a right child (denoted as `grand_child`), a step needs to be added in the right rotation: set `grand_child` as the left child of `node`.

![Right rotation with grand_child](avl_tree.assets/avltree_right_rotate_with_grandchild.png){ class="animation-figure" }

<p align="center"> Figure 7-27 &nbsp; Right rotation with grand_child </p>

"Right rotation" is a figurative term; in practice, it is achieved by modifying node pointers, as shown in the following code:

=== "Python"

    ```python title="avl_tree.py"
    def right_rotate(self, node: TreeNode | None) -> TreeNode | None:
        """右旋操作"""
        child = node.left
        grand_child = child.right
        # 以 child 为原点，将 node 向右旋转
        child.right = node
        node.left = grand_child
        # 更新节点高度
        self.update_height(node)
        self.update_height(child)
        # 返回旋转后子树的根节点
        return child
    ```

=== "C++"

    ```cpp title="avl_tree.cpp"
    /* 右旋操作 */
    TreeNode *rightRotate(TreeNode *node) {
        TreeNode *child = node->left;
        TreeNode *grandChild = child->right;
        // 以 child 为原点，将 node 向右旋转
        child->right = node;
        node->left = grandChild;
        // 更新节点高度
        updateHeight(node);
        updateHeight(child);
        // 返回旋转后子树的根节点
        return child;
    }
    ```

=== "Java"

    ```java title="avl_tree.java"
    /* 右旋操作 */
    TreeNode rightRotate(TreeNode node) {
        TreeNode child = node.left;
        TreeNode grandChild = child.right;
        // 以 child 为原点，将 node 向右旋转
        child.right = node;
        node.left = grandChild;
        // 更新节点高度
        updateHeight(node);
        updateHeight(child);
        // 返回旋转后子树的根节点
        return child;
    }
    ```

=== "C#"

    ```csharp title="avl_tree.cs"
    /* 右旋操作 */
    TreeNode? RightRotate(TreeNode? node) {
        TreeNode? child = node?.left;
        TreeNode? grandChild = child?.right;
        // 以 child 为原点，将 node 向右旋转
        child.right = node;
        node.left = grandChild;
        // 更新节点高度
        UpdateHeight(node);
        UpdateHeight(child);
        // 返回旋转后子树的根节点
        return child;
    }
    ```

=== "Go"

    ```go title="avl_tree.go"
    /* 右旋操作 */
    func (t *aVLTree) rightRotate(node *TreeNode) *TreeNode {
        child := node.Left
        grandChild := child.Right
        // 以 child 为原点，将 node 向右旋转
        child.Right = node
        node.Left = grandChild
        // 更新节点高度
        t.updateHeight(node)
        t.updateHeight(child)
        // 返回旋转后子树的根节点
        return child
    }
    ```

=== "Swift"

    ```swift title="avl_tree.swift"
    /* 右旋操作 */
    func rightRotate(node: TreeNode?) -> TreeNode? {
        let child = node?.left
        let grandChild = child?.right
        // 以 child 为原点，将 node 向右旋转
        child?.right = node
        node?.left = grandChild
        // 更新节点高度
        updateHeight(node: node)
        updateHeight(node: child)
        // 返回旋转后子树的根节点
        return child
    }
    ```

=== "JS"

    ```javascript title="avl_tree.js"
    /* 右旋操作 */
    #rightRotate(node) {
        const child = node.left;
        const grandChild = child.right;
        // 以 child 为原点，将 node 向右旋转
        child.right = node;
        node.left = grandChild;
        // 更新节点高度
        this.#updateHeight(node);
        this.#updateHeight(child);
        // 返回旋转后子树的根节点
        return child;
    }
    ```

=== "TS"

    ```typescript title="avl_tree.ts"
    /* 右旋操作 */
    rightRotate(node: TreeNode): TreeNode {
        const child = node.left;
        const grandChild = child.right;
        // 以 child 为原点，将 node 向右旋转
        child.right = node;
        node.left = grandChild;
        // 更新节点高度
        this.updateHeight(node);
        this.updateHeight(child);
        // 返回旋转后子树的根节点
        return child;
    }
    ```

=== "Dart"

    ```dart title="avl_tree.dart"
    /* 右旋操作 */
    TreeNode? rightRotate(TreeNode? node) {
      TreeNode? child = node!.left;
      TreeNode? grandChild = child!.right;
      // 以 child 为原点，将 node 向右旋转
      child.right = node;
      node.left = grandChild;
      // 更新节点高度
      updateHeight(node);
      updateHeight(child);
      // 返回旋转后子树的根节点
      return child;
    }
    ```

=== "Rust"

    ```rust title="avl_tree.rs"
    /* 右旋操作 */
    fn right_rotate(node: OptionTreeNodeRc) -> OptionTreeNodeRc {
        match node {
            Some(node) => {
                let child = node.borrow().left.clone().unwrap();
                let grand_child = child.borrow().right.clone();
                // 以 child 为原点，将 node 向右旋转
                child.borrow_mut().right = Some(node.clone());
                node.borrow_mut().left = grand_child;
                // 更新节点高度
                Self::update_height(Some(node));
                Self::update_height(Some(child.clone()));
                // 返回旋转后子树的根节点
                Some(child)
            }
            None => None,
        }
    }
    ```

=== "C"

    ```c title="avl_tree.c"
    /* 右旋操作 */
    TreeNode *rightRotate(TreeNode *node) {
        TreeNode *child, *grandChild;
        child = node->left;
        grandChild = child->right;
        // 以 child 为原点，将 node 向右旋转
        child->right = node;
        node->left = grandChild;
        // 更新节点高度
        updateHeight(node);
        updateHeight(child);
        // 返回旋转后子树的根节点
        return child;
    }
    ```

=== "Kotlin"

    ```kotlin title="avl_tree.kt"
    /* 右旋操作 */
    fun rightRotate(node: TreeNode?): TreeNode {
        val child = node!!.left
        val grandChild = child!!.right
        // 以 child 为原点，将 node 向右旋转
        child.right = node
        node.left = grandChild
        // 更新节点高度
        updateHeight(node)
        updateHeight(child)
        // 返回旋转后子树的根节点
        return child
    }
    ```

=== "Ruby"

    ```ruby title="avl_tree.rb"
    ### 右旋操作 ###
    def right_rotate(node)
      child = node.left
      grand_child = child.right
      # 以 child 为原点，将 node 向右旋转
      child.right = node
      node.left = grand_child
      # 更新节点高度
      update_height(node)
      update_height(child)
      # 返回旋转后子树的根节点
      child
    end
    ```

=== "Zig"

    ```zig title="avl_tree.zig"
    // 右旋操作
    fn rightRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {
        var child = node.?.left;
        var grandChild = child.?.right;
        // 以 child 为原点，将 node 向右旋转
        child.?.right = node;
        node.?.left = grandChild;
        // 更新节点高度
        self.updateHeight(node);
        self.updateHeight(child);
        // 返回旋转后子树的根节点
        return child;
    }
    ```

### 2. &nbsp; Left rotation

Correspondingly, if considering the "mirror" of the above unbalanced binary tree, the "left rotation" operation shown in Figure 7-28 needs to be performed.

![Left rotation operation](avl_tree.assets/avltree_left_rotate.png){ class="animation-figure" }

<p align="center"> Figure 7-28 &nbsp; Left rotation operation </p>

Similarly, as shown in Figure 7-29, when the `child` node has a left child (denoted as `grand_child`), a step needs to be added in the left rotation: set `grand_child` as the right child of `node`.

![Left rotation with grand_child](avl_tree.assets/avltree_left_rotate_with_grandchild.png){ class="animation-figure" }

<p align="center"> Figure 7-29 &nbsp; Left rotation with grand_child </p>

It can be observed that **the right and left rotation operations are logically symmetrical, and they solve two symmetrical types of imbalance**. Based on symmetry, by replacing all `left` with `right`, and all `right` with `left` in the implementation code of right rotation, we can get the implementation code for left rotation:

=== "Python"

    ```python title="avl_tree.py"
    def left_rotate(self, node: TreeNode | None) -> TreeNode | None:
        """左旋操作"""
        child = node.right
        grand_child = child.left
        # 以 child 为原点，将 node 向左旋转
        child.left = node
        node.right = grand_child
        # 更新节点高度
        self.update_height(node)
        self.update_height(child)
        # 返回旋转后子树的根节点
        return child
    ```

=== "C++"

    ```cpp title="avl_tree.cpp"
    /* 左旋操作 */
    TreeNode *leftRotate(TreeNode *node) {
        TreeNode *child = node->right;
        TreeNode *grandChild = child->left;
        // 以 child 为原点，将 node 向左旋转
        child->left = node;
        node->right = grandChild;
        // 更新节点高度
        updateHeight(node);
        updateHeight(child);
        // 返回旋转后子树的根节点
        return child;
    }
    ```

=== "Java"

    ```java title="avl_tree.java"
    /* 左旋操作 */
    TreeNode leftRotate(TreeNode node) {
        TreeNode child = node.right;
        TreeNode grandChild = child.left;
        // 以 child 为原点，将 node 向左旋转
        child.left = node;
        node.right = grandChild;
        // 更新节点高度
        updateHeight(node);
        updateHeight(child);
        // 返回旋转后子树的根节点
        return child;
    }
    ```

=== "C#"

    ```csharp title="avl_tree.cs"
    /* 左旋操作 */
    TreeNode? LeftRotate(TreeNode? node) {
        TreeNode? child = node?.right;
        TreeNode? grandChild = child?.left;
        // 以 child 为原点，将 node 向左旋转
        child.left = node;
        node.right = grandChild;
        // 更新节点高度
        UpdateHeight(node);
        UpdateHeight(child);
        // 返回旋转后子树的根节点
        return child;
    }
    ```

=== "Go"

    ```go title="avl_tree.go"
    /* 左旋操作 */
    func (t *aVLTree) leftRotate(node *TreeNode) *TreeNode {
        child := node.Right
        grandChild := child.Left
        // 以 child 为原点，将 node 向左旋转
        child.Left = node
        node.Right = grandChild
        // 更新节点高度
        t.updateHeight(node)
        t.updateHeight(child)
        // 返回旋转后子树的根节点
        return child
    }
    ```

=== "Swift"

    ```swift title="avl_tree.swift"
    /* 左旋操作 */
    func leftRotate(node: TreeNode?) -> TreeNode? {
        let child = node?.right
        let grandChild = child?.left
        // 以 child 为原点，将 node 向左旋转
        child?.left = node
        node?.right = grandChild
        // 更新节点高度
        updateHeight(node: node)
        updateHeight(node: child)
        // 返回旋转后子树的根节点
        return child
    }
    ```

=== "JS"

    ```javascript title="avl_tree.js"
    /* 左旋操作 */
    #leftRotate(node) {
        const child = node.right;
        const grandChild = child.left;
        // 以 child 为原点，将 node 向左旋转
        child.left = node;
        node.right = grandChild;
        // 更新节点高度
        this.#updateHeight(node);
        this.#updateHeight(child);
        // 返回旋转后子树的根节点
        return child;
    }
    ```

=== "TS"

    ```typescript title="avl_tree.ts"
    /* 左旋操作 */
    leftRotate(node: TreeNode): TreeNode {
        const child = node.right;
        const grandChild = child.left;
        // 以 child 为原点，将 node 向左旋转
        child.left = node;
        node.right = grandChild;
        // 更新节点高度
        this.updateHeight(node);
        this.updateHeight(child);
        // 返回旋转后子树的根节点
        return child;
    }
    ```

=== "Dart"

    ```dart title="avl_tree.dart"
    /* 左旋操作 */
    TreeNode? leftRotate(TreeNode? node) {
      TreeNode? child = node!.right;
      TreeNode? grandChild = child!.left;
      // 以 child 为原点，将 node 向左旋转
      child.left = node;
      node.right = grandChild;
      // 更新节点高度
      updateHeight(node);
      updateHeight(child);
      // 返回旋转后子树的根节点
      return child;
    }
    ```

=== "Rust"

    ```rust title="avl_tree.rs"
    /* 左旋操作 */
    fn left_rotate(node: OptionTreeNodeRc) -> OptionTreeNodeRc {
        match node {
            Some(node) => {
                let child = node.borrow().right.clone().unwrap();
                let grand_child = child.borrow().left.clone();
                // 以 child 为原点，将 node 向左旋转
                child.borrow_mut().left = Some(node.clone());
                node.borrow_mut().right = grand_child;
                // 更新节点高度
                Self::update_height(Some(node));
                Self::update_height(Some(child.clone()));
                // 返回旋转后子树的根节点
                Some(child)
            }
            None => None,
        }
    }
    ```

=== "C"

    ```c title="avl_tree.c"
    /* 左旋操作 */
    TreeNode *leftRotate(TreeNode *node) {
        TreeNode *child, *grandChild;
        child = node->right;
        grandChild = child->left;
        // 以 child 为原点，将 node 向左旋转
        child->left = node;
        node->right = grandChild;
        // 更新节点高度
        updateHeight(node);
        updateHeight(child);
        // 返回旋转后子树的根节点
        return child;
    }
    ```

=== "Kotlin"

    ```kotlin title="avl_tree.kt"
    /* 左旋操作 */
    fun leftRotate(node: TreeNode?): TreeNode {
        val child = node!!.right
        val grandChild = child!!.left
        // 以 child 为原点，将 node 向左旋转
        child.left = node
        node.right = grandChild
        // 更新节点高度
        updateHeight(node)
        updateHeight(child)
        // 返回旋转后子树的根节点
        return child
    }
    ```

=== "Ruby"

    ```ruby title="avl_tree.rb"
    ### 左旋操作 ###
    def left_rotate(node)
      child = node.right
      grand_child = child.left
      # 以 child 为原点，将 node 向左旋转
      child.left = node
      node.right = grand_child
      # 更新节点高度
      update_height(node)
      update_height(child)
      # 返回旋转后子树的根节点
      child
    end
    ```

=== "Zig"

    ```zig title="avl_tree.zig"
    // 左旋操作
    fn leftRotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {
        var child = node.?.right;
        var grandChild = child.?.left;
        // 以 child 为原点，将 node 向左旋转
        child.?.left = node;
        node.?.right = grandChild;
        // 更新节点高度
        self.updateHeight(node);
        self.updateHeight(child);
        // 返回旋转后子树的根节点
        return child;
    }
    ```

### 3. &nbsp; Right-left rotation

For the unbalanced node 3 shown in Figure 7-30, using either left or right rotation alone cannot restore balance to the subtree. In this case, a "left rotation" needs to be performed on `child` first, followed by a "right rotation" on `node`.

![Right-left rotation](avl_tree.assets/avltree_left_right_rotate.png){ class="animation-figure" }

<p align="center"> Figure 7-30 &nbsp; Right-left rotation </p>

### 4. &nbsp; Left-right rotation

As shown in Figure 7-31, for the mirror case of the above unbalanced binary tree, a "right rotation" needs to be performed on `child` first, followed by a "left rotation" on `node`.

![Left-right rotation](avl_tree.assets/avltree_right_left_rotate.png){ class="animation-figure" }

<p align="center"> Figure 7-31 &nbsp; Left-right rotation </p>

### 5. &nbsp; Choice of rotation

The four kinds of imbalances shown in Figure 7-32 correspond to the cases described above, respectively requiring right rotation, left-right rotation, right-left rotation, and left rotation.

![The four rotation cases of AVL tree](avl_tree.assets/avltree_rotation_cases.png){ class="animation-figure" }

<p align="center"> Figure 7-32 &nbsp; The four rotation cases of AVL tree </p>

As shown in Table 7-3, we determine which of the above cases an unbalanced node belongs to by judging the sign of the balance factor of the unbalanced node and its higher-side child's balance factor.

<p align="center"> Table 7-3 &nbsp; Conditions for Choosing Among the Four Rotation Cases </p>

<div class="center-table" markdown>

| Balance factor of unbalanced node | Balance factor of child node | Rotation method to use            |
| --------------------------------- | ---------------------------- | --------------------------------- |
| $> 1$ (Left-leaning tree)         | $\geq 0$                     | Right rotation                    |
| $> 1$ (Left-leaning tree)         | $<0$                         | Left rotation then right rotation |
| $< -1$ (Right-leaning tree)       | $\leq 0$                     | Left rotation                     |
| $< -1$ (Right-leaning tree)       | $>0$                         | Right rotation then left rotation |

</div>

For convenience, we encapsulate the rotation operations into a function. **With this function, we can perform rotations on various kinds of imbalances, restoring balance to unbalanced nodes**. The code is as follows:

=== "Python"

    ```python title="avl_tree.py"
    def rotate(self, node: TreeNode | None) -> TreeNode | None:
        """执行旋转操作，使该子树重新恢复平衡"""
        # 获取节点 node 的平衡因子
        balance_factor = self.balance_factor(node)
        # 左偏树
        if balance_factor > 1:
            if self.balance_factor(node.left) >= 0:
                # 右旋
                return self.right_rotate(node)
            else:
                # 先左旋后右旋
                node.left = self.left_rotate(node.left)
                return self.right_rotate(node)
        # 右偏树
        elif balance_factor < -1:
            if self.balance_factor(node.right) <= 0:
                # 左旋
                return self.left_rotate(node)
            else:
                # 先右旋后左旋
                node.right = self.right_rotate(node.right)
                return self.left_rotate(node)
        # 平衡树，无须旋转，直接返回
        return node
    ```

=== "C++"

    ```cpp title="avl_tree.cpp"
    /* 执行旋转操作，使该子树重新恢复平衡 */
    TreeNode *rotate(TreeNode *node) {
        // 获取节点 node 的平衡因子
        int _balanceFactor = balanceFactor(node);
        // 左偏树
        if (_balanceFactor > 1) {
            if (balanceFactor(node->left) >= 0) {
                // 右旋
                return rightRotate(node);
            } else {
                // 先左旋后右旋
                node->left = leftRotate(node->left);
                return rightRotate(node);
            }
        }
        // 右偏树
        if (_balanceFactor < -1) {
            if (balanceFactor(node->right) <= 0) {
                // 左旋
                return leftRotate(node);
            } else {
                // 先右旋后左旋
                node->right = rightRotate(node->right);
                return leftRotate(node);
            }
        }
        // 平衡树，无须旋转，直接返回
        return node;
    }
    ```

=== "Java"

    ```java title="avl_tree.java"
    /* 执行旋转操作，使该子树重新恢复平衡 */
    TreeNode rotate(TreeNode node) {
        // 获取节点 node 的平衡因子
        int balanceFactor = balanceFactor(node);
        // 左偏树
        if (balanceFactor > 1) {
            if (balanceFactor(node.left) >= 0) {
                // 右旋
                return rightRotate(node);
            } else {
                // 先左旋后右旋
                node.left = leftRotate(node.left);
                return rightRotate(node);
            }
        }
        // 右偏树
        if (balanceFactor < -1) {
            if (balanceFactor(node.right) <= 0) {
                // 左旋
                return leftRotate(node);
            } else {
                // 先右旋后左旋
                node.right = rightRotate(node.right);
                return leftRotate(node);
            }
        }
        // 平衡树，无须旋转，直接返回
        return node;
    }
    ```

=== "C#"

    ```csharp title="avl_tree.cs"
    /* 执行旋转操作，使该子树重新恢复平衡 */
    TreeNode? Rotate(TreeNode? node) {
        // 获取节点 node 的平衡因子
        int balanceFactorInt = BalanceFactor(node);
        // 左偏树
        if (balanceFactorInt > 1) {
            if (BalanceFactor(node?.left) >= 0) {
                // 右旋
                return RightRotate(node);
            } else {
                // 先左旋后右旋
                node!.left = LeftRotate(node!.left);
                return RightRotate(node);
            }
        }
        // 右偏树
        if (balanceFactorInt < -1) {
            if (BalanceFactor(node?.right) <= 0) {
                // 左旋
                return LeftRotate(node);
            } else {
                // 先右旋后左旋
                node!.right = RightRotate(node!.right);
                return LeftRotate(node);
            }
        }
        // 平衡树，无须旋转，直接返回
        return node;
    }
    ```

=== "Go"

    ```go title="avl_tree.go"
    /* 执行旋转操作，使该子树重新恢复平衡 */
    func (t *aVLTree) rotate(node *TreeNode) *TreeNode {
        // 获取节点 node 的平衡因子
        // Go 推荐短变量，这里 bf 指代 t.balanceFactor
        bf := t.balanceFactor(node)
        // 左偏树
        if bf > 1 {
            if t.balanceFactor(node.Left) >= 0 {
                // 右旋
                return t.rightRotate(node)
            } else {
                // 先左旋后右旋
                node.Left = t.leftRotate(node.Left)
                return t.rightRotate(node)
            }
        }
        // 右偏树
        if bf < -1 {
            if t.balanceFactor(node.Right) <= 0 {
                // 左旋
                return t.leftRotate(node)
            } else {
                // 先右旋后左旋
                node.Right = t.rightRotate(node.Right)
                return t.leftRotate(node)
            }
        }
        // 平衡树，无须旋转，直接返回
        return node
    }
    ```

=== "Swift"

    ```swift title="avl_tree.swift"
    /* 执行旋转操作，使该子树重新恢复平衡 */
    func rotate(node: TreeNode?) -> TreeNode? {
        // 获取节点 node 的平衡因子
        let balanceFactor = balanceFactor(node: node)
        // 左偏树
        if balanceFactor > 1 {
            if self.balanceFactor(node: node?.left) >= 0 {
                // 右旋
                return rightRotate(node: node)
            } else {
                // 先左旋后右旋
                node?.left = leftRotate(node: node?.left)
                return rightRotate(node: node)
            }
        }
        // 右偏树
        if balanceFactor < -1 {
            if self.balanceFactor(node: node?.right) <= 0 {
                // 左旋
                return leftRotate(node: node)
            } else {
                // 先右旋后左旋
                node?.right = rightRotate(node: node?.right)
                return leftRotate(node: node)
            }
        }
        // 平衡树，无须旋转，直接返回
        return node
    }
    ```

=== "JS"

    ```javascript title="avl_tree.js"
    /* 执行旋转操作，使该子树重新恢复平衡 */
    #rotate(node) {
        // 获取节点 node 的平衡因子
        const balanceFactor = this.balanceFactor(node);
        // 左偏树
        if (balanceFactor > 1) {
            if (this.balanceFactor(node.left) >= 0) {
                // 右旋
                return this.#rightRotate(node);
            } else {
                // 先左旋后右旋
                node.left = this.#leftRotate(node.left);
                return this.#rightRotate(node);
            }
        }
        // 右偏树
        if (balanceFactor < -1) {
            if (this.balanceFactor(node.right) <= 0) {
                // 左旋
                return this.#leftRotate(node);
            } else {
                // 先右旋后左旋
                node.right = this.#rightRotate(node.right);
                return this.#leftRotate(node);
            }
        }
        // 平衡树，无须旋转，直接返回
        return node;
    }
    ```

=== "TS"

    ```typescript title="avl_tree.ts"
    /* 执行旋转操作，使该子树重新恢复平衡 */
    rotate(node: TreeNode): TreeNode {
        // 获取节点 node 的平衡因子
        const balanceFactor = this.balanceFactor(node);
        // 左偏树
        if (balanceFactor > 1) {
            if (this.balanceFactor(node.left) >= 0) {
                // 右旋
                return this.rightRotate(node);
            } else {
                // 先左旋后右旋
                node.left = this.leftRotate(node.left);
                return this.rightRotate(node);
            }
        }
        // 右偏树
        if (balanceFactor < -1) {
            if (this.balanceFactor(node.right) <= 0) {
                // 左旋
                return this.leftRotate(node);
            } else {
                // 先右旋后左旋
                node.right = this.rightRotate(node.right);
                return this.leftRotate(node);
            }
        }
        // 平衡树，无须旋转，直接返回
        return node;
    }
    ```

=== "Dart"

    ```dart title="avl_tree.dart"
    /* 执行旋转操作，使该子树重新恢复平衡 */
    TreeNode? rotate(TreeNode? node) {
      // 获取节点 node 的平衡因子
      int factor = balanceFactor(node);
      // 左偏树
      if (factor > 1) {
        if (balanceFactor(node!.left) >= 0) {
          // 右旋
          return rightRotate(node);
        } else {
          // 先左旋后右旋
          node.left = leftRotate(node.left);
          return rightRotate(node);
        }
      }
      // 右偏树
      if (factor < -1) {
        if (balanceFactor(node!.right) <= 0) {
          // 左旋
          return leftRotate(node);
        } else {
          // 先右旋后左旋
          node.right = rightRotate(node.right);
          return leftRotate(node);
        }
      }
      // 平衡树，无须旋转，直接返回
      return node;
    }
    ```

=== "Rust"

    ```rust title="avl_tree.rs"
    /* 执行旋转操作，使该子树重新恢复平衡 */
    fn rotate(node: OptionTreeNodeRc) -> OptionTreeNodeRc {
        // 获取节点 node 的平衡因子
        let balance_factor = Self::balance_factor(node.clone());
        // 左偏树
        if balance_factor > 1 {
            let node = node.unwrap();
            if Self::balance_factor(node.borrow().left.clone()) >= 0 {
                // 右旋
                Self::right_rotate(Some(node))
            } else {
                // 先左旋后右旋
                let left = node.borrow().left.clone();
                node.borrow_mut().left = Self::left_rotate(left);
                Self::right_rotate(Some(node))
            }
        }
        // 右偏树
        else if balance_factor < -1 {
            let node = node.unwrap();
            if Self::balance_factor(node.borrow().right.clone()) <= 0 {
                // 左旋
                Self::left_rotate(Some(node))
            } else {
                // 先右旋后左旋
                let right = node.borrow().right.clone();
                node.borrow_mut().right = Self::right_rotate(right);
                Self::left_rotate(Some(node))
            }
        } else {
            // 平衡树，无须旋转，直接返回
            node
        }
    }
    ```

=== "C"

    ```c title="avl_tree.c"
    /* 执行旋转操作，使该子树重新恢复平衡 */
    TreeNode *rotate(TreeNode *node) {
        // 获取节点 node 的平衡因子
        int bf = balanceFactor(node);
        // 左偏树
        if (bf > 1) {
            if (balanceFactor(node->left) >= 0) {
                // 右旋
                return rightRotate(node);
            } else {
                // 先左旋后右旋
                node->left = leftRotate(node->left);
                return rightRotate(node);
            }
        }
        // 右偏树
        if (bf < -1) {
            if (balanceFactor(node->right) <= 0) {
                // 左旋
                return leftRotate(node);
            } else {
                // 先右旋后左旋
                node->right = rightRotate(node->right);
                return leftRotate(node);
            }
        }
        // 平衡树，无须旋转，直接返回
        return node;
    }
    ```

=== "Kotlin"

    ```kotlin title="avl_tree.kt"
    /* 执行旋转操作，使该子树重新恢复平衡 */
    fun rotate(node: TreeNode): TreeNode {
        // 获取节点 node 的平衡因子
        val balanceFactor = balanceFactor(node)
        // 左偏树
        if (balanceFactor > 1) {
            if (balanceFactor(node.left) >= 0) {
                // 右旋
                return rightRotate(node)
            } else {
                // 先左旋后右旋
                node.left = leftRotate(node.left)
                return rightRotate(node)
            }
        }
        // 右偏树
        if (balanceFactor < -1) {
            if (balanceFactor(node.right) <= 0) {
                // 左旋
                return leftRotate(node)
            } else {
                // 先右旋后左旋
                node.right = rightRotate(node.right)
                return leftRotate(node)
            }
        }
        // 平衡树，无须旋转，直接返回
        return node
    }
    ```

=== "Ruby"

    ```ruby title="avl_tree.rb"
    ### 执行旋转操作，使该子树重新恢复平衡 ###
    def rotate(node)
      # 获取节点 node 的平衡因子
      balance_factor = balance_factor(node)
      # 左遍树
      if balance_factor > 1
        if balance_factor(node.left) >= 0
          # 右旋
          return right_rotate(node)
        else
          # 先左旋后右旋
          node.left = left_rotate(node.left)
          return right_rotate(node)
        end
      # 右遍树
      elsif balance_factor < -1
        if balance_factor(node.right) <= 0
          # 左旋
          return left_rotate(node)
        else
          # 先右旋后左旋
          node.right = right_rotate(node.right)
          return left_rotate(node)
        end
      end
      # 平衡树，无须旋转，直接返回
      node
    end
    ```

=== "Zig"

    ```zig title="avl_tree.zig"
    // 执行旋转操作，使该子树重新恢复平衡
    fn rotate(self: *Self, node: ?*inc.TreeNode(T)) ?*inc.TreeNode(T) {
        // 获取节点 node 的平衡因子
        var balance_factor = self.balanceFactor(node);
        // 左偏树
        if (balance_factor > 1) {
            if (self.balanceFactor(node.?.left) >= 0) {
                // 右旋
                return self.rightRotate(node);
            } else {
                // 先左旋后右旋
                node.?.left = self.leftRotate(node.?.left);
                return self.rightRotate(node);
            }
        }
        // 右偏树
        if (balance_factor < -1) {
            if (self.balanceFactor(node.?.right) <= 0) {
                // 左旋
                return self.leftRotate(node);
            } else {
                // 先右旋后左旋
                node.?.right = self.rightRotate(node.?.right);
                return self.leftRotate(node);
            }
        }
        // 平衡树，无须旋转，直接返回
        return node;
    }
    ```

## 7.5.3 &nbsp; Common operations in AVL trees

### 1. &nbsp; Node insertion

The node insertion operation in AVL trees is similar to that in binary search trees. The only difference is that after inserting a node in an AVL tree, a series of unbalanced nodes may appear along the path from that node to the root node. Therefore, **we need to start from this node and perform rotation operations upwards to restore balance to all unbalanced nodes**. The code is as follows:

=== "Python"

    ```python title="avl_tree.py"
    def insert(self, val):
        """插入节点"""
        self._root = self.insert_helper(self._root, val)

    def insert_helper(self, node: TreeNode | None, val: int) -> TreeNode:
        """递归插入节点（辅助方法）"""
        if node is None:
            return TreeNode(val)
        # 1. 查找插入位置并插入节点
        if val < node.val:
            node.left = self.insert_helper(node.left, val)
        elif val > node.val:
            node.right = self.insert_helper(node.right, val)
        else:
            # 重复节点不插入，直接返回
            return node
        # 更新节点高度
        self.update_height(node)
        # 2. 执行旋转操作，使该子树重新恢复平衡
        return self.rotate(node)
    ```

=== "C++"

    ```cpp title="avl_tree.cpp"
    /* 插入节点 */
    void insert(int val) {
        root = insertHelper(root, val);
    }

    /* 递归插入节点（辅助方法） */
    TreeNode *insertHelper(TreeNode *node, int val) {
        if (node == nullptr)
            return new TreeNode(val);
        /* 1. 查找插入位置并插入节点 */
        if (val < node->val)
            node->left = insertHelper(node->left, val);
        else if (val > node->val)
            node->right = insertHelper(node->right, val);
        else
            return node;    // 重复节点不插入，直接返回
        updateHeight(node); // 更新节点高度
        /* 2. 执行旋转操作，使该子树重新恢复平衡 */
        node = rotate(node);
        // 返回子树的根节点
        return node;
    }
    ```

=== "Java"

    ```java title="avl_tree.java"
    /* 插入节点 */
    void insert(int val) {
        root = insertHelper(root, val);
    }

    /* 递归插入节点（辅助方法） */
    TreeNode insertHelper(TreeNode node, int val) {
        if (node == null)
            return new TreeNode(val);
        /* 1. 查找插入位置并插入节点 */
        if (val < node.val)
            node.left = insertHelper(node.left, val);
        else if (val > node.val)
            node.right = insertHelper(node.right, val);
        else
            return node; // 重复节点不插入，直接返回
        updateHeight(node); // 更新节点高度
        /* 2. 执行旋转操作，使该子树重新恢复平衡 */
        node = rotate(node);
        // 返回子树的根节点
        return node;
    }
    ```

=== "C#"

    ```csharp title="avl_tree.cs"
    /* 插入节点 */
    void Insert(int val) {
        root = InsertHelper(root, val);
    }

    /* 递归插入节点（辅助方法） */
    TreeNode? InsertHelper(TreeNode? node, int val) {
        if (node == null) return new TreeNode(val);
        /* 1. 查找插入位置并插入节点 */
        if (val < node.val)
            node.left = InsertHelper(node.left, val);
        else if (val > node.val)
            node.right = InsertHelper(node.right, val);
        else
            return node;     // 重复节点不插入，直接返回
        UpdateHeight(node);  // 更新节点高度
        /* 2. 执行旋转操作，使该子树重新恢复平衡 */
        node = Rotate(node);
        // 返回子树的根节点
        return node;
    }
    ```

=== "Go"

    ```go title="avl_tree.go"
    /* 插入节点 */
    func (t *aVLTree) insert(val int) {
        t.root = t.insertHelper(t.root, val)
    }

    /* 递归插入节点（辅助函数） */
    func (t *aVLTree) insertHelper(node *TreeNode, val int) *TreeNode {
        if node == nil {
            return NewTreeNode(val)
        }
        /* 1. 查找插入位置并插入节点 */
        if val < node.Val.(int) {
            node.Left = t.insertHelper(node.Left, val)
        } else if val > node.Val.(int) {
            node.Right = t.insertHelper(node.Right, val)
        } else {
            // 重复节点不插入，直接返回
            return node
        }
        // 更新节点高度
        t.updateHeight(node)
        /* 2. 执行旋转操作，使该子树重新恢复平衡 */
        node = t.rotate(node)
        // 返回子树的根节点
        return node
    }
    ```

=== "Swift"

    ```swift title="avl_tree.swift"
    /* 插入节点 */
    func insert(val: Int) {
        root = insertHelper(node: root, val: val)
    }

    /* 递归插入节点（辅助方法） */
    func insertHelper(node: TreeNode?, val: Int) -> TreeNode? {
        var node = node
        if node == nil {
            return TreeNode(x: val)
        }
        /* 1. 查找插入位置并插入节点 */
        if val < node!.val {
            node?.left = insertHelper(node: node?.left, val: val)
        } else if val > node!.val {
            node?.right = insertHelper(node: node?.right, val: val)
        } else {
            return node // 重复节点不插入，直接返回
        }
        updateHeight(node: node) // 更新节点高度
        /* 2. 执行旋转操作，使该子树重新恢复平衡 */
        node = rotate(node: node)
        // 返回子树的根节点
        return node
    }
    ```

=== "JS"

    ```javascript title="avl_tree.js"
    /* 插入节点 */
    insert(val) {
        this.root = this.#insertHelper(this.root, val);
    }

    /* 递归插入节点（辅助方法） */
    #insertHelper(node, val) {
        if (node === null) return new TreeNode(val);
        /* 1. 查找插入位置并插入节点 */
        if (val < node.val) node.left = this.#insertHelper(node.left, val);
        else if (val > node.val)
            node.right = this.#insertHelper(node.right, val);
        else return node; // 重复节点不插入，直接返回
        this.#updateHeight(node); // 更新节点高度
        /* 2. 执行旋转操作，使该子树重新恢复平衡 */
        node = this.#rotate(node);
        // 返回子树的根节点
        return node;
    }
    ```

=== "TS"

    ```typescript title="avl_tree.ts"
    /* 插入节点 */
    insert(val: number): void {
        this.root = this.insertHelper(this.root, val);
    }

    /* 递归插入节点（辅助方法） */
    insertHelper(node: TreeNode, val: number): TreeNode {
        if (node === null) return new TreeNode(val);
        /* 1. 查找插入位置并插入节点 */
        if (val < node.val) {
            node.left = this.insertHelper(node.left, val);
        } else if (val > node.val) {
            node.right = this.insertHelper(node.right, val);
        } else {
            return node; // 重复节点不插入，直接返回
        }
        this.updateHeight(node); // 更新节点高度
        /* 2. 执行旋转操作，使该子树重新恢复平衡 */
        node = this.rotate(node);
        // 返回子树的根节点
        return node;
    }
    ```

=== "Dart"

    ```dart title="avl_tree.dart"
    /* 插入节点 */
    void insert(int val) {
      root = insertHelper(root, val);
    }

    /* 递归插入节点（辅助方法） */
    TreeNode? insertHelper(TreeNode? node, int val) {
      if (node == null) return TreeNode(val);
      /* 1. 查找插入位置并插入节点 */
      if (val < node.val)
        node.left = insertHelper(node.left, val);
      else if (val > node.val)
        node.right = insertHelper(node.right, val);
      else
        return node; // 重复节点不插入，直接返回
      updateHeight(node); // 更新节点高度
      /* 2. 执行旋转操作，使该子树重新恢复平衡 */
      node = rotate(node);
      // 返回子树的根节点
      return node;
    }
    ```

=== "Rust"

    ```rust title="avl_tree.rs"
    /* 插入节点 */
    fn insert(&mut self, val: i32) {
        self.root = Self::insert_helper(self.root.clone(), val);
    }

    /* 递归插入节点（辅助方法） */
    fn insert_helper(node: OptionTreeNodeRc, val: i32) -> OptionTreeNodeRc {
        match node {
            Some(mut node) => {
                /* 1. 查找插入位置并插入节点 */
                match {
                    let node_val = node.borrow().val;
                    node_val
                }
                .cmp(&val)
                {
                    Ordering::Greater => {
                        let left = node.borrow().left.clone();
                        node.borrow_mut().left = Self::insert_helper(left, val);
                    }
                    Ordering::Less => {
                        let right = node.borrow().right.clone();
                        node.borrow_mut().right = Self::insert_helper(right, val);
                    }
                    Ordering::Equal => {
                        return Some(node); // 重复节点不插入，直接返回
                    }
                }
                Self::update_height(Some(node.clone())); // 更新节点高度

                /* 2. 执行旋转操作，使该子树重新恢复平衡 */
                node = Self::rotate(Some(node)).unwrap();
                // 返回子树的根节点
                Some(node)
            }
            None => Some(TreeNode::new(val)),
        }
    }
    ```

=== "C"

    ```c title="avl_tree.c"
    /* 插入节点 */
    void insert(AVLTree *tree, int val) {
        tree->root = insertHelper(tree->root, val);
    }

    /* 递归插入节点（辅助函数） */
    TreeNode *insertHelper(TreeNode *node, int val) {
        if (node == NULL) {
            return newTreeNode(val);
        }
        /* 1. 查找插入位置并插入节点 */
        if (val < node->val) {
            node->left = insertHelper(node->left, val);
        } else if (val > node->val) {
            node->right = insertHelper(node->right, val);
        } else {
            // 重复节点不插入，直接返回
            return node;
        }
        // 更新节点高度
        updateHeight(node);
        /* 2. 执行旋转操作，使该子树重新恢复平衡 */
        node = rotate(node);
        // 返回子树的根节点
        return node;
    }
    ```

=== "Kotlin"

    ```kotlin title="avl_tree.kt"
    /* 插入节点 */
    fun insert(_val: Int) {
        root = insertHelper(root, _val)
    }

    /* 递归插入节点（辅助方法） */
    fun insertHelper(n: TreeNode?, _val: Int): TreeNode {
        if (n == null)
            return TreeNode(_val)
        var node = n
        /* 1. 查找插入位置并插入节点 */
        if (_val < node._val)
            node.left = insertHelper(node.left, _val)
        else if (_val > node._val)
            node.right = insertHelper(node.right, _val)
        else
            return node // 重复节点不插入，直接返回
        updateHeight(node) // 更新节点高度
        /* 2. 执行旋转操作，使该子树重新恢复平衡 */
        node = rotate(node)
        // 返回子树的根节点
        return node
    }
    ```

=== "Ruby"

    ```ruby title="avl_tree.rb"
    ### 插入节点 ###
    def insert(val)
      @root = insert_helper(@root, val)
    end

    ### 递归插入节点（辅助方法）###
    def insert_helper(node, val)
      return TreeNode.new(val) if node.nil?
      # 1. 查找插入位置并插入节点
      if val < node.val
        node.left = insert_helper(node.left, val)
      elsif val > node.val
        node.right = insert_helper(node.right, val)
      else
        # 重复节点不插入，直接返回
        return node
      end
      # 更新节点高度
      update_height(node)
      # 2. 执行旋转操作，使该子树重新恢复平衡
      rotate(node)
    end
    ```

=== "Zig"

    ```zig title="avl_tree.zig"
    // 插入节点
    fn insert(self: *Self, val: T) !void {
        self.root = (try self.insertHelper(self.root, val)).?;
    }

    // 递归插入节点（辅助方法）
    fn insertHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) !?*inc.TreeNode(T) {
        var node = node_;
        if (node == null) {
            var tmp_node = try self.mem_allocator.create(inc.TreeNode(T));
            tmp_node.init(val);
            return tmp_node;
        }
        // 1. 查找插入位置并插入节点
        if (val < node.?.val) {
            node.?.left = try self.insertHelper(node.?.left, val);
        } else if (val > node.?.val) {
            node.?.right = try self.insertHelper(node.?.right, val);
        } else {
            return node;            // 重复节点不插入，直接返回
        }
        self.updateHeight(node);    // 更新节点高度
        // 2. 执行旋转操作，使该子树重新恢复平衡
        node = self.rotate(node);
        // 返回子树的根节点
        return node;
    }
    ```

### 2. &nbsp; Node removal

Similarly, based on the method of removing nodes in binary search trees, rotation operations need to be performed from the bottom up to restore balance to all unbalanced nodes. The code is as follows:

=== "Python"

    ```python title="avl_tree.py"
    def remove(self, val: int):
        """删除节点"""
        self._root = self.remove_helper(self._root, val)

    def remove_helper(self, node: TreeNode | None, val: int) -> TreeNode | None:
        """递归删除节点（辅助方法）"""
        if node is None:
            return None
        # 1. 查找节点并删除
        if val < node.val:
            node.left = self.remove_helper(node.left, val)
        elif val > node.val:
            node.right = self.remove_helper(node.right, val)
        else:
            if node.left is None or node.right is None:
                child = node.left or node.right
                # 子节点数量 = 0 ，直接删除 node 并返回
                if child is None:
                    return None
                # 子节点数量 = 1 ，直接删除 node
                else:
                    node = child
            else:
                # 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
                temp = node.right
                while temp.left is not None:
                    temp = temp.left
                node.right = self.remove_helper(node.right, temp.val)
                node.val = temp.val
        # 更新节点高度
        self.update_height(node)
        # 2. 执行旋转操作，使该子树重新恢复平衡
        return self.rotate(node)
    ```

=== "C++"

    ```cpp title="avl_tree.cpp"
    /* 删除节点 */
    void remove(int val) {
        root = removeHelper(root, val);
    }

    /* 递归删除节点（辅助方法） */
    TreeNode *removeHelper(TreeNode *node, int val) {
        if (node == nullptr)
            return nullptr;
        /* 1. 查找节点并删除 */
        if (val < node->val)
            node->left = removeHelper(node->left, val);
        else if (val > node->val)
            node->right = removeHelper(node->right, val);
        else {
            if (node->left == nullptr || node->right == nullptr) {
                TreeNode *child = node->left != nullptr ? node->left : node->right;
                // 子节点数量 = 0 ，直接删除 node 并返回
                if (child == nullptr) {
                    delete node;
                    return nullptr;
                }
                // 子节点数量 = 1 ，直接删除 node
                else {
                    delete node;
                    node = child;
                }
            } else {
                // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
                TreeNode *temp = node->right;
                while (temp->left != nullptr) {
                    temp = temp->left;
                }
                int tempVal = temp->val;
                node->right = removeHelper(node->right, temp->val);
                node->val = tempVal;
            }
        }
        updateHeight(node); // 更新节点高度
        /* 2. 执行旋转操作，使该子树重新恢复平衡 */
        node = rotate(node);
        // 返回子树的根节点
        return node;
    }
    ```

=== "Java"

    ```java title="avl_tree.java"
    /* 删除节点 */
    void remove(int val) {
        root = removeHelper(root, val);
    }

    /* 递归删除节点（辅助方法） */
    TreeNode removeHelper(TreeNode node, int val) {
        if (node == null)
            return null;
        /* 1. 查找节点并删除 */
        if (val < node.val)
            node.left = removeHelper(node.left, val);
        else if (val > node.val)
            node.right = removeHelper(node.right, val);
        else {
            if (node.left == null || node.right == null) {
                TreeNode child = node.left != null ? node.left : node.right;
                // 子节点数量 = 0 ，直接删除 node 并返回
                if (child == null)
                    return null;
                // 子节点数量 = 1 ，直接删除 node
                else
                    node = child;
            } else {
                // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
                TreeNode temp = node.right;
                while (temp.left != null) {
                    temp = temp.left;
                }
                node.right = removeHelper(node.right, temp.val);
                node.val = temp.val;
            }
        }
        updateHeight(node); // 更新节点高度
        /* 2. 执行旋转操作，使该子树重新恢复平衡 */
        node = rotate(node);
        // 返回子树的根节点
        return node;
    }
    ```

=== "C#"

    ```csharp title="avl_tree.cs"
    /* 删除节点 */
    void Remove(int val) {
        root = RemoveHelper(root, val);
    }

    /* 递归删除节点（辅助方法） */
    TreeNode? RemoveHelper(TreeNode? node, int val) {
        if (node == null) return null;
        /* 1. 查找节点并删除 */
        if (val < node.val)
            node.left = RemoveHelper(node.left, val);
        else if (val > node.val)
            node.right = RemoveHelper(node.right, val);
        else {
            if (node.left == null || node.right == null) {
                TreeNode? child = node.left ?? node.right;
                // 子节点数量 = 0 ，直接删除 node 并返回
                if (child == null)
                    return null;
                // 子节点数量 = 1 ，直接删除 node
                else
                    node = child;
            } else {
                // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
                TreeNode? temp = node.right;
                while (temp.left != null) {
                    temp = temp.left;
                }
                node.right = RemoveHelper(node.right, temp.val!.Value);
                node.val = temp.val;
            }
        }
        UpdateHeight(node);  // 更新节点高度
        /* 2. 执行旋转操作，使该子树重新恢复平衡 */
        node = Rotate(node);
        // 返回子树的根节点
        return node;
    }
    ```

=== "Go"

    ```go title="avl_tree.go"
    /* 删除节点 */
    func (t *aVLTree) remove(val int) {
        t.root = t.removeHelper(t.root, val)
    }

    /* 递归删除节点（辅助函数） */
    func (t *aVLTree) removeHelper(node *TreeNode, val int) *TreeNode {
        if node == nil {
            return nil
        }
        /* 1. 查找节点并删除 */
        if val < node.Val.(int) {
            node.Left = t.removeHelper(node.Left, val)
        } else if val > node.Val.(int) {
            node.Right = t.removeHelper(node.Right, val)
        } else {
            if node.Left == nil || node.Right == nil {
                child := node.Left
                if node.Right != nil {
                    child = node.Right
                }
                if child == nil {
                    // 子节点数量 = 0 ，直接删除 node 并返回
                    return nil
                } else {
                    // 子节点数量 = 1 ，直接删除 node
                    node = child
                }
            } else {
                // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
                temp := node.Right
                for temp.Left != nil {
                    temp = temp.Left
                }
                node.Right = t.removeHelper(node.Right, temp.Val.(int))
                node.Val = temp.Val
            }
        }
        // 更新节点高度
        t.updateHeight(node)
        /* 2. 执行旋转操作，使该子树重新恢复平衡 */
        node = t.rotate(node)
        // 返回子树的根节点
        return node
    }
    ```

=== "Swift"

    ```swift title="avl_tree.swift"
    /* 删除节点 */
    func remove(val: Int) {
        root = removeHelper(node: root, val: val)
    }

    /* 递归删除节点（辅助方法） */
    func removeHelper(node: TreeNode?, val: Int) -> TreeNode? {
        var node = node
        if node == nil {
            return nil
        }
        /* 1. 查找节点并删除 */
        if val < node!.val {
            node?.left = removeHelper(node: node?.left, val: val)
        } else if val > node!.val {
            node?.right = removeHelper(node: node?.right, val: val)
        } else {
            if node?.left == nil || node?.right == nil {
                let child = node?.left ?? node?.right
                // 子节点数量 = 0 ，直接删除 node 并返回
                if child == nil {
                    return nil
                }
                // 子节点数量 = 1 ，直接删除 node
                else {
                    node = child
                }
            } else {
                // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
                var temp = node?.right
                while temp?.left != nil {
                    temp = temp?.left
                }
                node?.right = removeHelper(node: node?.right, val: temp!.val)
                node?.val = temp!.val
            }
        }
        updateHeight(node: node) // 更新节点高度
        /* 2. 执行旋转操作，使该子树重新恢复平衡 */
        node = rotate(node: node)
        // 返回子树的根节点
        return node
    }
    ```

=== "JS"

    ```javascript title="avl_tree.js"
    /* 删除节点 */
    remove(val) {
        this.root = this.#removeHelper(this.root, val);
    }

    /* 递归删除节点（辅助方法） */
    #removeHelper(node, val) {
        if (node === null) return null;
        /* 1. 查找节点并删除 */
        if (val < node.val) node.left = this.#removeHelper(node.left, val);
        else if (val > node.val)
            node.right = this.#removeHelper(node.right, val);
        else {
            if (node.left === null || node.right === null) {
                const child = node.left !== null ? node.left : node.right;
                // 子节点数量 = 0 ，直接删除 node 并返回
                if (child === null) return null;
                // 子节点数量 = 1 ，直接删除 node
                else node = child;
            } else {
                // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
                let temp = node.right;
                while (temp.left !== null) {
                    temp = temp.left;
                }
                node.right = this.#removeHelper(node.right, temp.val);
                node.val = temp.val;
            }
        }
        this.#updateHeight(node); // 更新节点高度
        /* 2. 执行旋转操作，使该子树重新恢复平衡 */
        node = this.#rotate(node);
        // 返回子树的根节点
        return node;
    }
    ```

=== "TS"

    ```typescript title="avl_tree.ts"
    /* 删除节点 */
    remove(val: number): void {
        this.root = this.removeHelper(this.root, val);
    }

    /* 递归删除节点（辅助方法） */
    removeHelper(node: TreeNode, val: number): TreeNode {
        if (node === null) return null;
        /* 1. 查找节点并删除 */
        if (val < node.val) {
            node.left = this.removeHelper(node.left, val);
        } else if (val > node.val) {
            node.right = this.removeHelper(node.right, val);
        } else {
            if (node.left === null || node.right === null) {
                const child = node.left !== null ? node.left : node.right;
                // 子节点数量 = 0 ，直接删除 node 并返回
                if (child === null) {
                    return null;
                } else {
                    // 子节点数量 = 1 ，直接删除 node
                    node = child;
                }
            } else {
                // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
                let temp = node.right;
                while (temp.left !== null) {
                    temp = temp.left;
                }
                node.right = this.removeHelper(node.right, temp.val);
                node.val = temp.val;
            }
        }
        this.updateHeight(node); // 更新节点高度
        /* 2. 执行旋转操作，使该子树重新恢复平衡 */
        node = this.rotate(node);
        // 返回子树的根节点
        return node;
    }
    ```

=== "Dart"

    ```dart title="avl_tree.dart"
    /* 删除节点 */
    void remove(int val) {
      root = removeHelper(root, val);
    }

    /* 递归删除节点（辅助方法） */
    TreeNode? removeHelper(TreeNode? node, int val) {
      if (node == null) return null;
      /* 1. 查找节点并删除 */
      if (val < node.val)
        node.left = removeHelper(node.left, val);
      else if (val > node.val)
        node.right = removeHelper(node.right, val);
      else {
        if (node.left == null || node.right == null) {
          TreeNode? child = node.left ?? node.right;
          // 子节点数量 = 0 ，直接删除 node 并返回
          if (child == null)
            return null;
          // 子节点数量 = 1 ，直接删除 node
          else
            node = child;
        } else {
          // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
          TreeNode? temp = node.right;
          while (temp!.left != null) {
            temp = temp.left;
          }
          node.right = removeHelper(node.right, temp.val);
          node.val = temp.val;
        }
      }
      updateHeight(node); // 更新节点高度
      /* 2. 执行旋转操作，使该子树重新恢复平衡 */
      node = rotate(node);
      // 返回子树的根节点
      return node;
    }
    ```

=== "Rust"

    ```rust title="avl_tree.rs"
    /* 删除节点 */
    fn remove(&self, val: i32) {
        Self::remove_helper(self.root.clone(), val);
    }

    /* 递归删除节点（辅助方法） */
    fn remove_helper(node: OptionTreeNodeRc, val: i32) -> OptionTreeNodeRc {
        match node {
            Some(mut node) => {
                /* 1. 查找节点并删除 */
                if val < node.borrow().val {
                    let left = node.borrow().left.clone();
                    node.borrow_mut().left = Self::remove_helper(left, val);
                } else if val > node.borrow().val {
                    let right = node.borrow().right.clone();
                    node.borrow_mut().right = Self::remove_helper(right, val);
                } else if node.borrow().left.is_none() || node.borrow().right.is_none() {
                    let child = if node.borrow().left.is_some() {
                        node.borrow().left.clone()
                    } else {
                        node.borrow().right.clone()
                    };
                    match child {
                        // 子节点数量 = 0 ，直接删除 node 并返回
                        None => {
                            return None;
                        }
                        // 子节点数量 = 1 ，直接删除 node
                        Some(child) => node = child,
                    }
                } else {
                    // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
                    let mut temp = node.borrow().right.clone().unwrap();
                    loop {
                        let temp_left = temp.borrow().left.clone();
                        if temp_left.is_none() {
                            break;
                        }
                        temp = temp_left.unwrap();
                    }
                    let right = node.borrow().right.clone();
                    node.borrow_mut().right = Self::remove_helper(right, temp.borrow().val);
                    node.borrow_mut().val = temp.borrow().val;
                }
                Self::update_height(Some(node.clone())); // 更新节点高度

                /* 2. 执行旋转操作，使该子树重新恢复平衡 */
                node = Self::rotate(Some(node)).unwrap();
                // 返回子树的根节点
                Some(node)
            }
            None => None,
        }
    }
    ```

=== "C"

    ```c title="avl_tree.c"
    /* 删除节点 */
    // 由于引入了 stdio.h ，此处无法使用 remove 关键词
    void removeItem(AVLTree *tree, int val) {
        TreeNode *root = removeHelper(tree->root, val);
    }

    /* 递归删除节点（辅助函数） */
    TreeNode *removeHelper(TreeNode *node, int val) {
        TreeNode *child, *grandChild;
        if (node == NULL) {
            return NULL;
        }
        /* 1. 查找节点并删除 */
        if (val < node->val) {
            node->left = removeHelper(node->left, val);
        } else if (val > node->val) {
            node->right = removeHelper(node->right, val);
        } else {
            if (node->left == NULL || node->right == NULL) {
                child = node->left;
                if (node->right != NULL) {
                    child = node->right;
                }
                // 子节点数量 = 0 ，直接删除 node 并返回
                if (child == NULL) {
                    return NULL;
                } else {
                    // 子节点数量 = 1 ，直接删除 node
                    node = child;
                }
            } else {
                // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
                TreeNode *temp = node->right;
                while (temp->left != NULL) {
                    temp = temp->left;
                }
                int tempVal = temp->val;
                node->right = removeHelper(node->right, temp->val);
                node->val = tempVal;
            }
        }
        // 更新节点高度
        updateHeight(node);
        /* 2. 执行旋转操作，使该子树重新恢复平衡 */
        node = rotate(node);
        // 返回子树的根节点
        return node;
    }
    ```

=== "Kotlin"

    ```kotlin title="avl_tree.kt"
    /* 删除节点 */
    fun remove(_val: Int) {
        root = removeHelper(root, _val)
    }

    /* 递归删除节点（辅助方法） */
    fun removeHelper(n: TreeNode?, _val: Int): TreeNode? {
        var node = n ?: return null
        /* 1. 查找节点并删除 */
        if (_val < node._val)
            node.left = removeHelper(node.left, _val)
        else if (_val > node._val)
            node.right = removeHelper(node.right, _val)
        else {
            if (node.left == null || node.right == null) {
                val child = if (node.left != null)
                    node.left
                else
                    node.right
                // 子节点数量 = 0 ，直接删除 node 并返回
                if (child == null)
                    return null
                // 子节点数量 = 1 ，直接删除 node
                else
                    node = child
            } else {
                // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
                var temp = node.right
                while (temp!!.left != null) {
                    temp = temp.left
                }
                node.right = removeHelper(node.right, temp._val)
                node._val = temp._val
            }
        }
        updateHeight(node) // 更新节点高度
        /* 2. 执行旋转操作，使该子树重新恢复平衡 */
        node = rotate(node)
        // 返回子树的根节点
        return node
    }
    ```

=== "Ruby"

    ```ruby title="avl_tree.rb"
    ### 删除节点 ###
    def remove(val)
      @root = remove_helper(@root, val)
    end

    ### 递归删除节点（辅助方法）###
    def remove_helper(node, val)
      return if node.nil?
      # 1. 查找节点并删除
      if val < node.val
        node.left = remove_helper(node.left, val)
      elsif val > node.val
        node.right = remove_helper(node.right, val)
      else
        if node.left.nil? || node.right.nil?
          child = node.left || node.right
          # 子节点数量 = 0 ，直接删除 node 并返回
          return if child.nil?
          # 子节点数量 = 1 ，直接删除 node
          node = child
        else
          # 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
          temp = node.right
          while !temp.left.nil?
            temp = temp.left
          end
          node.right = remove_helper(node.right, temp.val)
          node.val = temp.val
        end
      end
      # 更新节点高度
      update_height(node)
      # 2. 执行旋转操作，使该子树重新恢复平衡
      rotate(node)
    end
    ```

=== "Zig"

    ```zig title="avl_tree.zig"
    // 删除节点
    fn remove(self: *Self, val: T) void {
       self.root = self.removeHelper(self.root, val).?;
    }

    // 递归删除节点（辅助方法）
    fn removeHelper(self: *Self, node_: ?*inc.TreeNode(T), val: T) ?*inc.TreeNode(T) {
        var node = node_;
        if (node == null) return null;
        // 1. 查找节点并删除
        if (val < node.?.val) {
            node.?.left = self.removeHelper(node.?.left, val);
        } else if (val > node.?.val) {
            node.?.right = self.removeHelper(node.?.right, val);
        } else {
            if (node.?.left == null or node.?.right == null) {
                var child = if (node.?.left != null) node.?.left else node.?.right;
                // 子节点数量 = 0 ，直接删除 node 并返回
                if (child == null) {
                    return null;
                // 子节点数量 = 1 ，直接删除 node
                } else {
                    node = child;
                }
            } else {
                // 子节点数量 = 2 ，则将中序遍历的下个节点删除，并用该节点替换当前节点
                var temp = node.?.right;
                while (temp.?.left != null) {
                    temp = temp.?.left;
                }
                node.?.right = self.removeHelper(node.?.right, temp.?.val);
                node.?.val = temp.?.val;
            }
        }
        self.updateHeight(node); // 更新节点高度
        // 2. 执行旋转操作，使该子树重新恢复平衡
        node = self.rotate(node);
        // 返回子树的根节点
        return node;
    }
    ```

### 3. &nbsp; Node search

The node search operation in AVL trees is consistent with that in binary search trees and will not be detailed here.

## 7.5.4 &nbsp; Typical applications of AVL trees

- Organizing and storing large amounts of data, suitable for scenarios with high-frequency searches and low-frequency intertions and removals.
- Used to build index systems in databases.
- Red-black trees are also a common type of balanced binary search tree. Compared to AVL trees, red-black trees have more relaxed balancing conditions, require fewer rotations for node insertion and removal, and have a higher average efficiency for node addition and removal operations.