1.. 平衡二叉树
  • 平衡二叉树要求,对于任意一个节点,左子树和右子树的高度差不能超过1。
  • 平衡二叉树的高度和节点数量之间的关系也是O(logn)
  • 为二叉树标注节点高度并计算平衡因子
  • 第三十二篇 玩转数据结构——AVL树(AVL Tree)-LMLPHP
  • AVL树是一棵平衡二叉树

2.. 实现AVL树的业务逻辑

  • import java.util.ArrayList;
    
    public class AVLTree<K extends Comparable<K>, V> {
    
        private class Node {
    public K key;
    public V value;
    public Node left;
    public Node right;
    public int height; // 构造函数
    public Node(K key, V value) {
    this.key = key;
    this.value = value;
    left = null;
    right = null;
    height = 1;
    }
    } private Node root;
    private int size; // 构造函数
    public AVLTree() {
    root = null;
    size = 0;
    } // 实现getSize方法
    public int getSize() {
    return size;
    } // 实现isEmpty方法
    public boolean isEmpty() {
    return size == 0;
    } // 判断该二叉树是否为二分搜索树
    public boolean isBST() {
    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;
    } private void inOrder(Node node, ArrayList<K> keys) { if (node == null) {
    return;
    }
    inOrder(node.left, keys);
    keys.add(node.key);
    inOrder(node.right, keys);
    } // 判断二叉树是否为平衡二叉树
    public boolean isBalanced() {
    return isBalanced(root);
    } // 判断以node为根的二叉树是否为平衡二叉树
    private boolean isBalanced(Node node) { if (node == null) {
    return true;
    }
    int balanceFactor = getBalanceFactor(node);
    if (Math.abs(balanceFactor) > 1) {
    return false;
    }
    return isBalanced(node.left) && isBalanced(node.right);
    } // 返回节点node的高度值
    private int getHeight(Node node) {
    if (node == null) {
    return 0;
    }
    return node.height;
    } // 返回节点node的平衡因子
    private int getBalanceFactor(Node node) {
    if (node == null) {
    return 0;
    }
    return getHeight(node.left) - getHeight(node.right);
    } // 对节点y进行向右旋转操作,返回旋转后新的根节点x
    // y x
    // / \ / \
    // x T4 向右旋转 (y) z y
    // / \ - - - - - - - -> / \ / \
    // z T3 T1 T2 T3 T4
    // / \
    // T1 T2
    private Node rightRotate(Node y) { Node x = y.left;
    Node T3 = x.right; // 向右旋转
    x.right = y;
    y.left = T3; // 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;
    } // 对节点y进行向左旋转操作,返回旋转后新的根节点x
    // y x
    // / \ / \
    // T1 x 向左旋转 (y) y z
    // / \ - - - - - - - -> / \ / \
    // T2 z T1 T2 T3 T4
    // / \
    // T3 T4
    private Node leftRotate(Node y) { Node x = y.right;
    Node T2 = x.left; // 向左旋转
    x.left = y;
    y.right = T2; //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;
    } // 实现add方法
    public void add(K key, V value) {
    root = add(root, key, value);
    } // 向以node为根节点的二分搜索树中插入元素(key, value),递归算法
    // 返回插入新元素后的二分搜索树的根
    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) {
    node.right = add(node.right, key, value);
    } else {
    node.value = value;
    } // 更新height值
    node.height = 1 + Math.max(getHeight(node.left), getHeight(node.right)); // 计算平衡因子
    int balanceFactor = getBalanceFactor(node); // 平衡维护
    // 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) {
    node.left = leftRotate(node.left);
    return rightRotate(node);
    }
    // RL
    if (balanceFactor < -1 && getBalanceFactor(node.right) > 0) {
    node.right = rightRotate(node.right);
    return leftRotate(node);
    } return node;
    } // 返回以node为根节点的二分搜索树中,key所在的节点
    private Node getNode(Node node, K key) { if (node == null)
    return null; if (key.compareTo(node.key) < 0) {
    return getNode(node.left, key);
    } else if (key.compareTo(node.key) > 0) {
    return getNode(node.right, key);
    } else {
    return node;
    }
    } 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 newValue) {
    Node node = getNode(root, key);
    if (node == null)
    throw new IllegalArgumentException(key + " doesn't exist!"); node.value = newValue;
    } // 返回以node为根的二分搜索树的最小元素所在节点
    private Node minimum(Node node) {
    if (node.left == null) {
    return node;
    }
    return minimum(node.left);
    } // 实现remove方法
    // 删除二分搜索树中键为key的节点
    public V remove(K key) {
    Node node = getNode(root, key); if (node != null) {
    root = remove(root, key);
    return node.value;
    }
    return null;
    } // 删除以node为根节点的二分搜索树中键为key的节点,递归算法
    // 返回删除节点后新的二分搜索树的根
    private Node remove(Node node, K key) {
    if (node == null) {
    return null;
    } Node retNode;
    if (key.compareTo(node.key) < 0) {
    node.left = remove(node.left, key);
    retNode = node;
    } else if (key.compareTo(node.key) > 0) {
    node.right = remove(node.right, key);
    retNode = node;
    } else {
    // 待删除节点左子树为空的情况
    if (node.left == null) {
    Node rightNode = node.right;
    node.right = null;
    size--;
    retNode = rightNode;
    // 待删除节点右子树为空的情况
    } else if (node.right == null) {
    Node leftNode = node.left;
    node.left = null;
    size--;
    retNode = leftNode;
    // 待删除节点左右子树均不为空
    // 找到比待删除节点大的最小节点,即待删除节点右子树的最小节点
    // 用这个节点顶替待删除节点
    } else {
    Node successor = minimum(node.right);
    successor.right = remove(node.right, successor.key); //这里进行了size--操作
    successor.left = node.left;
    node.left = null;
    node.right = null;
    retNode = successor;
    }
    } if (retNode == null) {
    return null;
    } // 更新height值
    retNode.height = 1 + Math.max(getHeight(retNode.left), getHeight(retNode.right)); // 计算平衡因子
    int balanceFactor = getBalanceFactor(retNode); // 平衡维护
    // LL
    if (balanceFactor > 1 && getBalanceFactor(retNode.left) >= 0) {
    return rightRotate(retNode);
    }
    // RR
    if (balanceFactor < -1 && getBalanceFactor(retNode.right) <= 0) {
    return leftRotate(retNode);
    } // LR
    if (balanceFactor > 1 && getBalanceFactor(retNode.left) < 0) {
    node.left = leftRotate(retNode.left);
    return rightRotate(retNode);
    }
    // RL
    if (balanceFactor < -1 && getBalanceFactor(retNode.right) > 0) {
    node.right = rightRotate(retNode.right);
    return leftRotate(retNode);
    } return retNode;
    } // 打印测试
    public static void main(String[] args) { System.out.println("Pride and Prejudice"); ArrayList<String> words = new ArrayList<>(); if (FileOperation.readFile("pride-and-prejudice.txt", words)) { System.out.println("Total words: " + words.size()); AVLTree<String, Integer> map = new AVLTree<>();
    for (String word : words) {
    if (map.contains(word)) {
    map.set(word, map.get(word) + 1);
    } else {
    map.add(word, 1);
    }
    } System.out.println("Total different words: " + map.getSize());
    System.out.println("Frequency of PRIDE: " + map.get("pride"));
    System.out.println("Frequency of PREJUDICE: " + map.get("prejudice")); System.out.println("is BST: " + map.isBST()); System.out.println("is Balanced: " + map.isBalanced());
    }
    }
    }
05-08 15:35