數據結構學習之平衡樹與AVL篇(Java)

平衡樹：對於任意一個節點，左子樹和右子樹的高度差不能超過1
平衡因子：任意一個節點左子樹與右子樹的高度差.
AVL樹是平衡二叉樹的一種，AVL樹本身首先是一棵二叉搜索樹。因爲二分搜索樹當插入的數據比較有規律時，二分搜索樹最差的情況可能會退化爲一個鏈表。就失去了使用這種數據結構來處理數據的意義。所以AVL這種自平衡二叉查找樹就最先被髮明出來了。AVL樹也被稱爲高度平衡樹，增加和刪除可能需要通過一次或多次樹旋轉來重新平衡這個樹。
使用java實現AVLTree:

import java.util.ArrayList;

/**
 * @author ymn
 * @version 1.0
 * @date 2020\6\2 0002
 */
public class AVLTree<K extends Comparable<K>,V> {

     private class Node{
         public K key;
         public V value;
         public Node left,right;
         public int height;

         public Node(K key,V value){
           this.key = key;
           this.value = value;
           left = null;
           right = null;
           height = 1;
         }
    }

    private Node root;
    private int size;

    public AVLTree(){
        root = null;
        size = 0;
    }

    public int getSize(){
        return size;
    }
    public boolean isEmpty(){
        return size == 0;
    }
    //判斷該二叉樹是否是一顆二分搜索樹
    public boolean isBST(){
        //利用二分搜索樹中序遍歷的結果是二分搜索樹中數據從小到大排列的結果
        ArrayList<K> keys = new ArrayList<>();
        inOrder(root,keys);
        for (int i = 1;i < keys.size(); i++){
            if (keys.get(i - 1).compareTo(keys.get(i)) > 0){
                return false;
            }
        }
        return true;
    }
    //中序遍歷
    private void inOrder(Node node,ArrayList<K> keys){
        if (node == null)
            return;
        inOrder(node.left,keys);
        keys.add(node.key);
        inOrder(node.right,keys);
    }

    //判斷該二叉樹是否是一顆平衡二叉樹
    public boolean isBalanced(){
        return isBalanced(root);
    }
    //判斷以node爲根的二叉樹是否是平衡二叉樹，遞歸算法
    private boolean isBalanced(Node node){
        if (node == null){
            return true;
        }
        int balanceFactor = getBalanceFactor(node);
        //Math.abs爲取絕對值
        if (Math.abs(balanceFactor) > 1){
            return false;
        }
        return isBalanced(node.left) && isBalanced(node.right);
    }

    //對節點y進行右旋轉操作，返回旋轉後新的節點x
    private Node rightRotate(Node y){
        Node x = y.left;
        Node xRight = x.right;
        //向右旋轉過程
        x.right = y;
        y.left = xRight;
        //更新height
        y.height = Math.max(getHeight(y.left),getHeight(y.right)) + 1;
        x.height = Math.max(getHeight(x.left),getHeight(x.right)) + 1;
        return x;
    }
    //對節點y進行左旋轉操作，返回旋轉後新的節點x
    private Node leftRotate(Node y){
        Node x = y.right;
        Node xLeft = x.left;
        //向左旋轉處理
        x.left = y;
        y.left = xLeft;
        //更新height
        y.height = Math.max(getHeight(y.left),getHeight(y.right)) + 1;
        x.height = Math.max(getHeight(x.left),getHeight(x.right)) + 1;
        return x;
    }
    //獲取節點的高度
    private int getHeight(Node node){
        if (node == null){
            return 0;
        }
        return node.height;
    }
    //獲得節點的平衡因子
    private int getBalanceFactor(Node node){
      if (node == null){
          return 0;
      }
      return getHeight(node.left) - getHeight(node.right);
    }

    public void add(K key,V value){
        root = add(root,key,value);
    }

    //向以node爲根的二分搜索樹中插入元素(key,value),遞歸算法
    //返回插入新節點後二分搜索樹的根
    private Node add(Node node,K key,V value){
        if(node == null){
            size ++;
            return new Node(key, value);
        }
        if (key.compareTo(node.key) < 0){
            node.left = add(node.left,key, value);
        }else if(key.compareTo(node.key) > 0){ //插入重複元素說明什麼也不做
            node.right = add(node.right,key, value);
        }else {
            node.value = value;
        }
        //更新height
        node.height = 1 + Math.max(getHeight(node.left),getHeight(node.right));
        //計算平衡因子
        int balanceFactor = getBalanceFactor(node);
//        if (Math.abs(balanceFactor) > 1){
//            System.out.println("unbalanced" + balanceFactor);
//        }
        //平衡維護
        //LL
        if (balanceFactor > 1 && getBalanceFactor(node.left) >= 0)
            return rightRotate(node);
        //RR
        if (balanceFactor < -1 && getBalanceFactor(node.right) <= 0)
            return leftRotate(node);
        //LR
        if(balanceFactor > 1 && getBalanceFactor(node.left) < 0) {
            //先對不平衡節點的左孩子進行左旋轉
            node.left = leftRotate(node.left);
            //在進行一次右旋轉
            return rightRotate(node);
        }
        //RL
        if (balanceFactor < -1 && getBalanceFactor(node.right) > 0){
            //先對不平衡節點的右孩子進行右旋轉
            node.right = rightRotate(node.right);
            //在進行一次左旋轉
            return leftRotate(node);
        }
        return node;
    }

    //返回以node爲根節點的二分搜索樹中，key所在的節點
    private Node getNode(Node node,K key){
        if (node == null){
            return null;
        }
        if (key.compareTo(node.key) == 0){
            return node;
        }else if (key.compareTo(node.key) < 0){
            return getNode(node.left,key);
        }else {   //key.compareTo(node.key) > 0
            return getNode(node.right,key);
        }
    }

    public boolean contains(K key){
        return getNode(root,key) != null;
    }

    public V get(K key){
        Node node = getNode(root,key);
        return node == null ? null : node.value;
    }

    public void set(K key, V value) {
        Node node = getNode(root,key);
        if (node == null){
            throw new IllegalArgumentException(key + "dose't exist!");
        }
        node.value = value;
    }

    //返回以node爲根的二分搜索樹的最小值所在的節點,遞歸算法
    private Node minimum(Node node){
        if (node.left == null){
            return node;
        }
        return minimum(node.left);
    }

    public V remove(K key){
        Node node = getNode(root,key);
        if (node != null){
            root = remove(root,key);
            return node.value;
        }
        return null;
    }

    //刪除掉以node爲根的二分搜索樹中值爲e節點，遞歸算法
    //返回刪除節點後新的二分搜索樹的根
    private Node remove(Node node,K key){
        if (node == null){
            return null;
        }
        Node returnNode;
        if (key.compareTo(node.key) < 0){
            node.left = remove(node.left,key);
            returnNode = node;
        }else if (key.compareTo(node.key) > 0){
            node.right = remove(node.right,key);
            returnNode = node;
        }
        else { //key == node.key
            //待刪除節點左子樹爲空的情況
            if (node.left == null){
                Node rightNode = node.right;
                node.right = null;
                size --;
                returnNode = rightNode;
            }
            //待刪除節點右子樹爲空的情況
            else if (node.right == null){
                Node leftNode = node.left;
                node.left = null;
                size --;
                returnNode = leftNode;
            }
            else {
                //待刪除節點均不爲空的情況
                //找到比待刪除節點大的最小節點,即待刪除節點右子樹的最小節點(後繼節點)。
                //或找到比待刪除節點小的最大節點，即待刪除節點左子樹的最大節點(前驅節點）
                //用這個節點頂替待刪除節點的位置
                Node successor = minimum(node.right);
                successor.right = remove(node.right, successor.key);
                successor.left = node.left;
                node.left = null;
                node.right = null;
                returnNode = successor;
            }
        }
        //如果刪除節點後returnNode爲空，比如刪除的是葉子節點
        if (returnNode == null){
            return null;
        }
        //更新height
        returnNode.height = 1 + Math.max(getHeight(returnNode.left),getHeight(returnNode.right));
        //計算平衡因子
        int balanceFactor = getBalanceFactor(returnNode);
        //平衡維護
        //LL
        if (balanceFactor > 1 && getBalanceFactor(returnNode.left) >= 0)
            return rightRotate(returnNode);
        //RR
        if (balanceFactor < -1 && getBalanceFactor(returnNode.right) <= 0)
            return leftRotate(returnNode);
        //LR
        if(balanceFactor > 1 && getBalanceFactor(returnNode.left) < 0) {
            //先對不平衡節點的左孩子進行左旋轉
            returnNode.left = leftRotate(returnNode.left);
            //在進行一次右旋轉
            return rightRotate(returnNode);
        }
        //RL
        if (balanceFactor < -1 && getBalanceFactor(returnNode.right) > 0){
            //先對不平衡節點的右孩子進行右旋轉
            returnNode.right = rightRotate(returnNode.right);
            //在進行一次左旋轉
            return leftRotate(returnNode);
        }
        return returnNode;
    }
}