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
