Java data structure AVL tree example analysis

Introduction of AVL tree

Searching a binary tree has extremely high search efficiency, but searching a binary tree will result in the following extreme situations:

Like this The search efficiency of binary tree is even lower than that of linked list. The balanced binary tree (AVL tree) that appears based on the search binary tree solves this problem. When the absolute value of the height difference between the left and right subtrees of a node in a balanced binary tree (AVL tree) is greater than 1, their height difference will be reduced through a rotation operation.

Basic concept

The AVL tree is essentially a binary search tree. Its characteristics are:

1. itself is first a binary search tree.

2. The absolute value (balance factor) of the difference between the heights of the left and right subtrees of each node is at most 1. In other words, the AVL tree is essentially a binary search tree (binary sorting tree, binary search tree) with balancing function.

3. When inserting a node or deleting a node, the absolute value of the height difference between the left and right subtrees of a node is greater than 1. In this case, you need to pass Left rotation and Right rotation operation makes the binary tree reach a balanced state again.

Balance Factor (balanceFactor)

• The left subtree and right subtree of a node The height difference is .

• The BF of any node in the AVL tree can only be -1, 0 and 1.

Basic design

The following are the simple methods and attributes required by the AVL tree:

public class AVLTree <E extends Comparable<E>>{
    class Node{
        E value;
        Node left;
        Node right;
        int height;
        public Node(){}
        public Node(E value){
            this.value = value;
            height = 1;
            left = null;
            right = null;
        }
        public void display(){
            System.out.print(this.value + " ");
        }
    }
    Node root;
    int size;
    public int size(){
        return size;
    }
    public int getHeight(Node node) {
        if(node == null) return 0;
        return node.height;
    }
    //获取平衡因子(左右子树的高度差，大小为1或者0是平衡的，大小大于1不平衡)
    public int getBalanceFactor(){
        return getBalanceFactor(root);
    }
    public int getBalanceFactor(Node node){
        if(node == null) return 0;
        return getHeight(node.left) - getHeight(node.right);
    }

    //判断一个树是否是一个平衡二叉树
    public boolean isBalance(Node node){
        if(node == null) return true;
        int balanceFactor = Math.abs(getBalanceFactor(node.left) - getBalanceFactor(node.right));
        if(balanceFactor > 1) return false;
        return isBalance(node.left) && isBalance(node.right);
    }
    public boolean isBalance(){
        return isBalance(root);
    }

    //中序遍历树
    private  void inPrevOrder(Node root){
        if(root == null) return;
        inPrevOrder(root.left);
        root.display();
        inPrevOrder(root.right);
    }
    public void inPrevOrder(){
        System.out.print("中序遍历：");
        inPrevOrder(root);
    }}

RR(Left-handed)

Insert a node into the right subtree of the right subtree of a tree, causing the binary tree to become unbalanced, as shown below. Insert 5 into the balanced binary tree, causing the tree to become unbalanced. At this time, a left-turn operation is required, as follows:

The code is as follows:

//左旋,并且返回新的根节点
    public Node leftRotate(Node node){
        System.out.println("leftRotate");
       Node cur = node.right;
       node.right = cur.left;
       cur.left = node;
       //跟新node和cur的高度
        node.height = Math.max(getHeight(node.left),getHeight(node.right)) + 1;
        cur.height = Math.max(getHeight(cur.left),getHeight(cur.right)) + 1;
        return cur;
    }

LL (right rotation)

Insert a node into the left subtree of the left subtree of an AVL tree, causing the binary tree to become It is not balanced, as shown below. Insert 2 into the balanced binary tree, causing the tree to become unbalanced. At this time, a left-turn operation is required, as follows:

The code is as follows:

 //右旋，并且返回新的根节点
    public Node rightRotate(Node node){
        System.out.println("rightRotate");
        Node cur = node.left;
        node.left = cur.right;
        cur.right = node;
        //跟新node和cur的高度
        node.height = Math.max(getHeight(node.left),getHeight(node.right)) + 1;
        cur.height = Math.max(getHeight(cur.left),getHeight(cur.right)) + 1;
        return cur;
    }

LR (First rotate left and then rotate right)

Insert a node into the right subtree of the left subtree of the AVL tree. As a result, the tree is no longer balanced. You need to first correct the left subtree. Perform left rotation , and then rotate the entire tree right , as shown in the figure below, insert node 5.

RL (first rotate right and then rotate left)

Insert a node into the left subtree of the AVL treeright subtree, causing the tree to no longer be balanced. You need to right-rotate the right subtree first. Then left-rotate the entire tree, as shown in the figure below, insert node 2.

Add node

//添加元素
    public  void add(E e){
        root = add(root,e);
    }
    public Node add(Node node, E value) {
        if (node == null) {
            size++;
            return new Node(value);
        }
        if (value.compareTo(node.value) > 0) {
            node.right = add(node.right, value);
        } else if (value.compareTo(node.value) < 0) {
            node.left = add(node.left, value);
        }
        //跟新节点高度
        node.height = Math.max(getHeight(node.left), getHeight(node.right)) + 1;
        //获取当前节点的平衡因子
        int balanceFactor = getBalanceFactor(node);
        //该子树不平衡且新插入节点(导致不平衡的节点)在左子树的左子树上，此时需要进行右旋
        if (balanceFactor > 1 && getBalanceFactor(node.left) >= 0) {
            return rightRotate(node);
        }
        //该子树不平衡且新插入节点(导致不平衡的节点)在右子树子树的右子树上，此时需要进行左旋
        else if (balanceFactor < -1 && getBalanceFactor(node.right) <= 0) {
            return leftRotate(node);
        }
        //该子树不平衡且新插入节点(导致不平衡的节点)在左子树的右子树上，此时需要先对左子树左旋，在整个树右旋
        else if (balanceFactor > 1 && getBalanceFactor(node.left) < 0) {
            node.left = leftRotate(node.left);
            return rightRotate(node);
        }
        //balanceFactor < -1 && getBalanceFactor(node.left) > 0
        //该子树不平衡且新插入节点(导致不平衡的节点)在右子树的左子树上，此时需要先对右子树右旋，再整个树左旋
        else if(balanceFactor < -1 && getBalanceFactor(node.right) > 0) {
            node.right = rightRotate(node.right);
            return leftRotate(node);
        }
        return node;
    }

Delete node

 //删除节点
    public E remove(E value){
        root = remove(root,value);
        if(root == null){
            return null;
        }
        return root.value;
    }
    public Node remove(Node node, E value){
        Node retNode = null;
        if(node == null)
            return retNode;
        if(value.compareTo(node.value) > 0){
            node.right = remove(node.right,value);
            retNode = node;
        }
        else if(value.compareTo(node.value) < 0){
            node.left = remove(node.left,value);
            retNode = node;
        }
        //value.compareTo(node.value) = 0
        else{
            //左右节点都为空，或者左节点为空
            if(node.left == null){
                size--;
                retNode = node.right;
            }
            //右节点为空
            else if(node.right == null){
                size--;
                retNode = node.left;
            }
            //左右节点都不为空
            else{
                Node successor = new Node();
                //寻找右子树最小的节点
                Node cur = node.right;
                while(cur.left != null){
                    cur = cur.left;
                }
                successor.value  = cur.value;
                successor.right = remove(node.right,value);
                successor.left = node.left;
                node.left =  node.right = null;
                retNode = successor;
            }
            if(retNode == null)
                return null;
            //维护二叉树平衡
            //跟新height
            retNode.height = Math.max(getHeight(retNode.left),getHeight(retNode.right));
        }
        int balanceFactor = getBalanceFactor(retNode);
        //该子树不平衡且新插入节点(导致不平衡的节点)在左子树的左子树上，此时需要进行右旋
        if (balanceFactor > 1 && getBalanceFactor(retNode.left) >= 0) {
            return rightRotate(retNode);
        }
        //该子树不平衡且新插入节点(导致不平衡的节点)在右子树子树的右子树上，此时需要进行左旋
        else if (balanceFactor < -1 && getBalanceFactor(retNode.right) <= 0) {
            return leftRotate(retNode);
        }
        //该子树不平衡且新插入节点(导致不平衡的节点)在左子树的右子树上，此时需要先对左子树左旋，在整个树右旋
        else if (balanceFactor > 1 && getBalanceFactor(retNode.left) < 0) {
            retNode.left = leftRotate(retNode.left);
            return rightRotate(retNode);
        }
        //该子树不平衡且新插入节点(导致不平衡的节点)在右子树的左子树上，此时需要先对右子树右旋，再整个树左旋
        else if(balanceFactor < -1 && getBalanceFactor(retNode.right) > 0) {
            retNode.right = rightRotate(retNode.right);
            return leftRotate(retNode);
        }
        return  retNode;
    }

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