AVL Trees

AVL trees are a type of self-balancing binary search tree. They were the first self-balancing trees to be invented, with the initials AVL standing for the creators, Adelson-Velsky and Landis.

Self-Balancing

The key feature of AVL trees is that they always maintain balance. A balanced tree is a tree where the heights of the left and right subtrees of any node differ by at most 1.

Maintaining balance is important because it ensures that the time complexity of basic operations remains efficient. In an unbalanced binary search tree, operations like search, insert, and delete can take longer to perform, reducing the efficiency of the tree.

Height-Balanced

AVL trees are height-balanced, meaning that the heights of the left and right subtrees of any node differ by at most 1. This balance is achieved through a process called rotation.

When a new node is inserted, the AVL tree checks for any imbalances and performs rotations to restore balance. There are four possible rotations: left rotation, right rotation, left-right rotation, and right-left rotation.

Advantages of AVL Trees

AVL trees offer several advantages:

• Efficient searching, insertion, and deletion operations with a guaranteed time complexity of O(log n)
• Automatic balancing ensures optimal performance
• Well-suited for scenarios where the tree is frequently modified

Example

Let's take a look at an example of an AVL tree:

1      30
2     /  \
3    20   40
4   / \     \
5  10  25    50

In this example, the AVL tree has a height of 2, and the left and right subtrees of the root node have a height difference of 1. This is an example of a balanced AVL tree.

AVL trees are commonly used in various applications, such as in database systems, as indexes in search algorithms, and in in-memory data structures.

To implement an AVL tree, we need to define a class that represents a node in the tree and another class that represents the AVL tree itself. Here's an implementation of the Node and AVLTree classes in Java:

1// Node class
2class Node {
3    int data;
4    int height;
5    Node left;
6    Node right;
7
8    public Node(int data) {
9        this.data = data;
10        this.height = 1;
11        this.left = null;
12        this.right = null;
13    }
14}
15
16// AVL Tree class
17class AVLTree {
18    Node root;
19
20    // Insert method, rotation methods, and other functionality
21}

In the implementation above, the Node class represents a node in the AVL tree, while the AVLTree class represents the tree itself. The Node class stores the data, height, and references to the left and right child nodes.

To maintain balance, the AVLTree class performs rotations when necessary during the insertion of new nodes. These rotations ensure that the AVL tree remains height-balanced and efficient for searching, insertion, and deletion operations.

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}

import java.util.LinkedList;

import java.util.Queue;

​

// Node class

class Node {

    int data;

    int height;

    Node left;

    Node right;

​

    // Constructor

class Node {

    int data;

    int height;

    Node left;

    Node right;

​

    // Constructor

class Node {

    int data;

    int height;

    Node left;

    Node right;

​

    // Constructor

class Node {

    int data;

    int height;

    Node left;