【转载】图解：二叉搜索树算法（BST）

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“岁月极美，在于它必然的流逝”
“春花秋月夏日冬雪”
— 三毛

一、树 & 二叉树

树是由节点和边构成，储存元素的集合。节点分根节点、父节点和子节点的概念。
如图：树深=4; 5是根节点；同样8与3的关系是父子节点关系。

二叉树binary tree，则加了“二叉”（binary），意思是在树中作区分。每个节点至多有两个子（child）,left child & right child。二叉树在很多例子中使用，比如二叉树表示算术表达式。
如图：1/8是左节点；2/3是右节点；

二、二叉搜索树 BST

顾名思义，二叉树上又加了个搜索的限制。其要求：每个节点比其左子树元素大，比其右子树元素小。
如图：每个节点比它左子树的任意节点大，而且比它右子树的任意节点小

三、BST Java实现

直接上代码，对应代码分享在 Github 主页
BinarySearchTree.java

package org.algorithm.tree;
/*
 * Copyright [2015] [Jeff Lee]
 *
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 *   http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */
 
/**
 * 二叉搜索树(BST)实现
 *
 * Created by bysocket on 16/7/7.
 */
public class BinarySearchTree {
    /**
     * 根节点
     */
    public static TreeNode root;
 
    public BinarySearchTree() {
        this.root = null;
    }
 
    /**
     * 查找
     *      树深(N) O(lgN)
     *      1. 从root节点开始
     *      2. 比当前节点值小,则找其左节点
     *      3. 比当前节点值大,则找其右节点
     *      4. 与当前节点值相等,查找到返回TRUE
     *      5. 查找完毕未找到,
     * @param key
     * @return
     */
    public TreeNode search (int key) {
        TreeNode current = root;
        while (current != null
                && key != current.value) {
            if (key < current.value )
                current = current.left;
            else
                current = current.right;
        }
        return current;
    }
 
    /**
     * 插入
     *      1. 从root节点开始
     *      2. 如果root为空,root为插入值
     *      循环:
     *      3. 如果当前节点值大于插入值,找左节点
     *      4. 如果当前节点值小于插入值,找右节点
     * @param key
     * @return
     */
    public TreeNode insert (int key) {
        // 新增节点
        TreeNode newNode = new TreeNode(key);
        // 当前节点
        TreeNode current = root;
        // 上个节点
        TreeNode parent  = null;
        // 如果根节点为空
        if (current == null) {
            root = newNode;
            return newNode;
        }
        while (true) {
            parent = current;
            if (key < current.value) {
                current = current.left;
                if (current == null) {
                    parent.left = newNode;
                    return newNode;
                }
            } else {
                current = current.right;
                if (current == null) {
                    parent.right = newNode;
                    return newNode;
                }
            }
        }
    }
 
    /**
     * 删除节点
     *      1.找到删除节点
     *      2.如果删除节点左节点为空 , 右节点也为空;
     *      3.如果删除节点只有一个子节点 右节点 或者 左节点
     *      4.如果删除节点左右子节点都不为空
     * @param key
     * @return
     */
    public TreeNode delete (int key) {
        TreeNode parent  = root;
        TreeNode current = root;
        boolean isLeftChild = false;
        // 找到删除节点 及 是否在左子树
        while (current.value != key) {
            parent = current;
            if (current.value > key) {
                isLeftChild = true;
                current = current.left;
            } else {
                isLeftChild = false;
                current = current.right;
            }
 
            if (current == null) {
                return current;
            }
        }
 
        // 如果删除节点左节点为空 , 右节点也为空
        if (current.left == null && current.right == null) {
            if (current == root) {
                root = null;
            }
            // 在左子树
            if (isLeftChild == true) {
                parent.left = null;
            } else {
                parent.right = null;
            }
        }
        // 如果删除节点只有一个子节点 右节点 或者 左节点
        else if (current.right == null) {
            if (current == root) {
                root = current.left;
            } else if (isLeftChild) {
                parent.left = current.left;
            } else {
                parent.right = current.left;
            }
 
        }
        else if (current.left == null) {
            if (current == root) {
                root = current.right;
            } else if (isLeftChild) {
                parent.left = current.right;
            } else {
                parent.right = current.right;
            }
        }
        // 如果删除节点左右子节点都不为空
        else if (current.left != null && current.right != null) {
            // 找到删除节点的后继者
            TreeNode successor = getDeleteSuccessor(current);
            if (current == root) {
                root = successor;
            } else if (isLeftChild) {
                parent.left = successor;
            } else {
                parent.right = successor;
            }
            successor.left = current.left;
        }
        return current;
    }
 
    /**
     * 获取删除节点的后继者
     *      删除节点的后继者是在其右节点树种最小的节点
     * @param deleteNode
     * @return
     */
    public TreeNode getDeleteSuccessor(TreeNode deleteNode) {
        // 后继者
        TreeNode successor = null;
        TreeNode successorParent = null;
        TreeNode current = deleteNode.right;
 
        while (current != null) {
            successorParent = successor;
            successor = current;
            current = current.left;
        }
 
        // 检查后继者(不可能有左节点树)是否有右节点树
        // 如果它有右节点树,则替换后继者位置,加到后继者父亲节点的左节点.
        if (successor != deleteNode.right) {
            successorParent.left = successor.right;
            successor.right = deleteNode.right;
        }
 
        return successor;
    }
 
    public void toString(TreeNode root) {
        if (root != null) {
            toString(root.left);
            System.out.print("value = " + root.value + " -> ");
            toString(root.right);
        }
    }
}
 
/**
 * 节点
 */
class TreeNode {
 
    /**
     * 节点值
     */
    int value;
 
    /**
     * 左节点
     */
    TreeNode left;
 
    /**
     * 右节点
     */
    TreeNode right;
 
    public TreeNode(int value) {
        this.value = value;
        left  = null;
        right = null;
    }
}

1. 节点数据结构
首先定义了节点的数据接口，节点分左节点和右节点及本身节点值。如图

代码如下：

/**
 * 节点
 */
class TreeNode {
 
    /**
     * 节点值
     */
    int value;
 
    /**
     * 左节点
     */
    TreeNode left;
 
    /**
     * 右节点
     */
    TreeNode right;
 
    public TreeNode(int value) {
        this.value = value;
        left  = null;
        right = null;
    }
}

2. 插入
插入，和删除一样会引起二叉搜索树的动态变化。插入相对删处理逻辑相对简单些。如图插入的逻辑：

a. 从root节点开始
b.如果root为空,root为插入值
c.循环:
d.如果当前节点值大于插入值,找左节点
e.如果当前节点值小于插入值,找右节点
代码对应：

/**
 * 插入
 *      1. 从root节点开始
 *      2. 如果root为空,root为插入值
 *      循环:
 *      3. 如果当前节点值大于插入值,找左节点
 *      4. 如果当前节点值小于插入值,找右节点
 * @param key
 * @return
 */
public TreeNode insert (int key) {
    // 新增节点
    TreeNode newNode = new TreeNode(key);
    // 当前节点
    TreeNode current = root;
    // 上个节点
    TreeNode parent  = null;
    // 如果根节点为空
    if (current == null) {
        root = newNode;
        return newNode;
    }
    while (true) {
        parent = current;
        if (key < current.value) {
            current = current.left;
            if (current == null) {
                parent.left = newNode;
                return newNode;
            }
        } else {
            current = current.right;
            if (current == null) {
                parent.right = newNode;
                return newNode;
            }
        }
    }
}

3.查找

其算法复杂度 : O(lgN),树深(N)。如图查找逻辑：

a.从root节点开始
b.比当前节点值小,则找其左节点
c.比当前节点值大,则找其右节点
d.与当前节点值相等,查找到返回TRUE
e.查找完毕未找到
代码对应：

/**
 * 查找
 *      树深(N) O(lgN)
 *      1. 从root节点开始
 *      2. 比当前节点值小,则找其左节点
 *      3. 比当前节点值大,则找其右节点
 *      4. 与当前节点值相等,查找到返回TRUE
 *      5. 查找完毕未找到,
 * @param key
 * @return
 */
public TreeNode search (int key) {
    TreeNode current = root;
    while (current != null
            && key != current.value) {
        if (key < current.value )
            current = current.left;
        else
            current = current.right;
    }
    return current;
}

4. 删除
首先找到删除节点，其寻找方法：删除节点的后继者是在其右节点树种最小的节点。如图删除对应逻辑：

结果为：

a.找到删除节点
b.如果删除节点左节点为空 , 右节点也为空;
c.如果删除节点只有一个子节点右节点或者左节点
d.如果删除节点左右子节点都不为空
代码对应见上面完整代码。

案例测试代码如下，BinarySearchTreeTest.java

package org.algorithm.tree;
/*
 * Copyright [2015] [Jeff Lee]
 *
 * Licensed under the Apache License, Version 2.0 (the "License");
 * you may not use this file except in compliance with the License.
 * You may obtain a copy of the License at
 *
 *   http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */
 
/**
 * 二叉搜索树(BST)测试案例 {@link BinarySearchTree}
 *
 * Created by bysocket on 16/7/10.
 */
public class BinarySearchTreeTest {
 
    public static void main(String[] args) {
        BinarySearchTree b = new BinarySearchTree();
        b.insert(3);b.insert(8);b.insert(1);b.insert(4);b.insert(6);
        b.insert(2);b.insert(10);b.insert(9);b.insert(20);b.insert(25);
 
        // 打印二叉树
        b.toString(b.root);
        System.out.println();
 
        // 是否存在节点值10
        TreeNode node01 = b.search(10);
        System.out.println("是否存在节点值为10 => " + node01.value);
        // 是否存在节点值11
        TreeNode node02 = b.search(11);
        System.out.println("是否存在节点值为11 => " + node02);
 
        // 删除节点8
        TreeNode node03 = b.delete(8);
        System.out.println("删除节点8 => " + node03.value);
        b.toString(b.root);
 
 
    }
}

运行结果如下：

value = 1 -> value = 2 -> value = 3 -> value = 4 -> value = 6 -> value = 8 -> value = 9 -> value = 10 -> value = 20 -> value = 25 -> 
是否存在节点值为10 => 10
是否存在节点值为11 => null
删除节点8 => 8
value = 1 -> value = 2 -> value = 3 -> value = 4 -> value = 6 -> value = 9 -> value = 10 -> value = 20 -> value = 25 ->

四、小结

与偶尔吃一碗“老坛酸菜牛肉面”一样的味道，品味一个算法，比如BST，的时候，总是那种说不出的味道。

树，二叉树的概念

BST算法