二叉搜索树(二叉排序树)

一.概念

二叉搜索树又称二叉排序树，具有以下性质：

• 若它的左子树不为空，则左子树上所有节点的值都小于根节点的值
• 若它的右子树不为空，则右子树上所有节点的值都大于根节点的值
• 它的左右子树也分别为二叉搜索树

注意：二叉搜索树中序遍历的结果是有序的

二、基本操作

1.查找元素

思路：二叉搜索树的左子树永远是比根节点小的，而它的右子树则都是比根节点大的值。当前节点比要找的大就往左走，当前元素比要找的小就往右走

 public Node search(int key) { 
    if(root == null) { 
    return null; } Node cur = root; while (cur != null) { 
    if(cur.val == key) { 
    return cur; }else if(cur.val > key) { 
    cur = cur.left; }else{ 
    cur = cur.right; } } return null; } 

2.插入元素

代码实现：

 public boolean insert(int key) { 
    Node node = new Node(key); if(root == null) { 
    this.root = node; return true; } Node parent = null; Node cur = root; while (cur != null) { 
    if(cur.val == key) { 
    //有相同的元素直接return return false; }else if(cur.val > key) { 
    parent = cur; cur = cur.left; }else{ 
    parent = cur; cur = cur.right; } } if (parent.val > key) { 
    parent.left = node; }else{ 
    parent.right = node; } return true; } 

3.删除元素

删除元素是一个比较难的点，要考虑到很多种情况

1. cur.left == null
   1. cur 是 root，则 root = cur.right
   2. cur 不是 root，cur 是 parent.left，则 parent.left = cur.right
   3. cur 不是 root，cur 是 parent.right，则 parent.right = cur.right
2. cur.right == null
   1. cur 是 root，则 root = cur.left
   2. cur 不是 root，cur 是 parent.left，则 parent.left = cur.left
   3. cur 不是 root，cur 是 parent.right，则 parent.right = cur.left
3. cur.left != null && cur.right != null
   采用替罪羊的方式删除
   
   
   1. 找到要删除节点，右树最左边的节点或者找到左树最右边的节点，替换这个两个节点的val值。
   2. 这样才能保证，删除后左树一定比根节点小，右树一定比根节点大
      
      
      

 public boolean remove(int key) { 
    if(this.root == null) { 
    return false; } Node parent = null; Node cur = this.root; while (cur != null) { 
    if(cur.val == key) { 
    removeKey(parent,cur); return true; }else if(cur.val < key) { 
    parent = cur; cur = cur.right; }else{ 
    parent = cur; cur = cur.left; } } return false; } public void removeKey(Node parent,Node cur) { 
    if(cur.left == null) { 
    if(cur == this.root) { 
    this.root = this.root.right; }else if(cur == parent.left) { 
    parent.left = cur.right; }else{ 
    parent.right = cur.right; } }else if(cur.right == null) { 
    if(this.root == cur) { 
    this.root = this.root.left; }else if(cur == parent.left) { 
    parent.left = cur.left; }else{ 
    parent.right = cur.left; } }else{ 
   //左右都不为空的情况 Node targetParent = cur; Node target = cur.right; while (target.left != null) { 
    targetParent = target; target = target.left; } cur.val = target.val; if(targetParent.left == target) { 
    targetParent.left = target.right; }else{ 
    targetParent.right = target.right; } } } 

4.性能分析

最好情况：二叉搜索树为完全二叉树，其平均比较次数为 O(log${}_2$n)

最坏情况：二叉搜索树退化为单支树，其平均比较次数为：O

所有代码：

public class BinarySearchTree { 
    public static class Node { 
    int val; Node left; Node right; public Node(int val) { 
    this.val = val; } } public Node root = null; / * 查找某个节点 * @param key */ public Node search(int key) { 
    if(root == null) { 
    return null; } Node cur = root; while (cur != null) { 
    if(cur.val == key) { 
    return cur; }else if(cur.val > key) { 
    cur = cur.left; }else{ 
    cur = cur.right; } } return null; } / * 插入元素 * @param key * @return */ public boolean insert(int key) { 
    Node node = new Node(key); if(root == null) { 
    this.root = node; return true; } Node parent = null; Node cur = root; while (cur != null) { 
    if(cur.val == key) { 
    //有相同的元素直接return return false; }else if(cur.val > key) { 
    parent = cur; cur = cur.left; }else{ 
    parent = cur; cur = cur.right; } } if (parent.val > key) { 
    parent.left = node; }else{ 
    parent.right = node; } return true; } / * 删除元素 * @param key */ public boolean remove(int key) { 
    if(this.root == null) { 
    return false; } Node parent = null; Node cur = this.root; while (cur != null) { 
    if(cur.val == key) { 
    removeKey(parent,cur); return true; }else if(cur.val < key) { 
    parent = cur; cur = cur.right; }else{ 
    parent = cur; cur = cur.left; } } return false; } public void removeKey(Node parent,Node cur) { 
    if(cur.left == null) { 
    if(cur == this.root) { 
    this.root = this.root.right; }else if(cur == parent.left) { 
    parent.left = cur.right; }else{ 
    parent.right = cur.right; } }else if(cur.right == null) { 
    if(this.root == cur) { 
    this.root = this.root.left; }else if(cur == parent.left) { 
    parent.left = cur.left; }else{ 
    parent.right = cur.left; } }else{ 
    Node targetParent = cur; Node target = cur.right; while (target.left != null) { 
    targetParent = target; target = target.left; } cur.val = target.val; if(targetParent.left == target) { 
    targetParent.left = target.right; }else{ 
    targetParent.right = target.right; } } } }