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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 the figure below, after two node removal operations, this binary search tree will degrade into a linked list.

Degradation of an AVL tree after removing nodes

For example, in the perfect binary tree shown in the figure below, 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

In 1962, G. M. Adelson-Velsky and E. M. Landis proposed the AVL Tree 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.

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 balanced binary search tree.

Node height

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

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

/* 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){}
};

/* 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; }
}

/* 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
}

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

/* 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
    }
}

/* 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;
    }
}

/* 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; 
    }
}

/* 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]);
}

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
        }))
    }
}

/* 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;
}

/* 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
}

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:

[file]{avl_tree}-[class]{avl_tree}-[func]{update_height}

Node balance factor

The balance factor 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:

[file]{avl_tree}-[class]{avl_tree}-[func]{balance_factor}

Tip

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

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.

Right rotation

As shown in the figure below, 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.

Steps of right rotation

avltree_right_rotate_step2

avltree_right_rotate_step3

avltree_right_rotate_step4

As shown in the figure below, 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

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

[file]{avl_tree}-[class]{avl_tree}-[func]{right_rotate}

Left rotation

Correspondingly, if considering the "mirror" of the above unbalanced binary tree, the "left rotation" operation shown in the figure below needs to be performed.

Left rotation operation

Similarly, as shown in the figure below, 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

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:

[file]{avl_tree}-[class]{avl_tree}-[func]{left_rotate}

Left-right rotation

For the unbalanced node 3 shown in the figure below, 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.

Left-right rotation

Right-left rotation

As shown in the figure below, 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.

Right-left rotation

Choice of rotation

The four kinds of imbalances shown in the figure below 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

As shown in the table below, 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.

Table   Conditions for Choosing Among the Four Rotation Cases

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

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:

[file]{avl_tree}-[class]{avl_tree}-[func]{rotate}

Common operations in AVL trees

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:

[file]{avl_tree}-[class]{avl_tree}-[func]{insert_helper}

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:

[file]{avl_tree}-[class]{avl_tree}-[func]{remove_helper}

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

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.