• 平衡二叉樹要求，對於任意一個節點，左子樹和右子樹的高度差不能超過1。
• 平衡二叉樹的高度和節點數量之間的關系也是O(logn)
• 為二叉樹標註節點高度並計算平衡因子
• 

• AVL樹是一棵平衡二叉樹

2.. 實現AVL樹的業務邏輯

• 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;
  
          // 構造函數
          public Node(K key, V value) {
              this.key = key;
              this.value = value;
              left = null;
              right = null;
              height = 1;
          }
      }
  
      private Node root;
      
   
  private int size;
  
      // 構造函數
      public AVLTree() {
          root = null;
          size = 0;
      }
  
      // 實現getSize方法
      public int getSize() {
          return size;
      }
  
      // 實現isEmpty方法
      public boolean isEmpty() {
          return size == 0;
      }
  
      // 判斷該二叉樹是否為二分搜索樹
      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);
      }
  
      // 判斷二叉樹是否為平衡二叉樹
      public boolean isBalanced() {
          return isBalanced(root);
      }
  
      // 判斷以node為根的二叉樹是否為平衡二叉樹
      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);
      }
  
      // 返回節點node的高度值
      private int getHeight(Node node) {
          if (node == null) {
              return 0;
          }
          return node.height;
      }
  
      // 返回節點node的平衡因子
      private int getBalanceFactor(Node node) {
          if (node == null) {
              return 0;
          }
          return getHeight(node.left) - getHeight(node.right);
      }
  
      // 對節點y進行向右旋轉操作，返回旋轉後新的根節點x
      //        y                              x
      //       / \                           /       //      x   T4     向右旋轉 (y)        z     y
      //     / \       - - - - - - - ->    / \   /     //    z   T3                       T1  T2 T3 T4
      //   /     // T1   T2
      private Node rightRotate(Node y) {
  
          Node x = y.left;
          Node T3 = x.right;
  
          // 向右旋轉
          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;
      }
  
      // 對節點y進行向左旋轉操作，返回旋轉後新的根節點x
      //    y                             x
      //  /  \                          /       // T1   x      向左旋轉 (y)       y     z
      //     / \   - - - - - - - ->   / \   /     //   T2  z                     T1 T2 T3 T4
      //      /     //     T3 T4
      private Node leftRotate(Node y) {
  
          Node x = y.right;
          Node T2 = x.left;
  
          // 向左旋轉
          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;
      }
  
      // 實現add方法
      public void add(K key, V value) {
          root = add(root, key, value);
      }
  
      // 向以node為根節點的二分搜索樹中插入元素(key, value)，遞歸算法
      // 返回插入新元素後的二分搜索樹的根
      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) {
              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));
  
          // 計算平衡因子
          int balanceFactor = getBalanceFactor(node);
  
          // 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) {
              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為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 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為根的二分搜索樹的最小元素所在節點
      private Node minimum(Node node) {
          if (node.left == null) {
              return node;
          }
          return minimum(node.left);
      }
  
      // 實現remove方法
      // 刪除二分搜索樹中鍵為key的節點
      public V remove(K key) {
          Node node = getNode(root, key);
  
          if (node != null) {
              root = remove(root, key);
              return node.value;
          }
          return null;
      }
  
      // 刪除以node為根節點的二分搜索樹中鍵為key的節點，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 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 {
              // 待刪除節點左子樹為空的情況
              if (node.left == null) {
                  Node rightNode = node.right;
                  node.right = null;
                  size--;
                  retNode = rightNode;
                  // 待刪除節點右子樹為空的情況
              } else if (node.right == null) {
                  Node leftNode = node.left;
                  node.left = null;
                  size--;
                  retNode = leftNode;
                  // 待刪除節點左右子樹均不為空
                  // 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
                  // 用這個節點頂替待刪除節點
              } else {
                  Node successor = minimum(node.right);
                  successor.right = remove(node.right, successor.key);  //這裏進行了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));
  
          // 計算平衡因子
          int balanceFactor = getBalanceFactor(retNode);
  
          // 平衡維護
          // 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;
      }
  
      // 打印測試
      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());
          }
      }
  }

第三十二篇 玩轉數據結構——AVL樹