Java實現 二叉搜尋樹演算法（BST）
阿新 • • 發佈：2019-02-18
一、樹 & 二叉樹
樹是由節點和邊構成，儲存元素的集合。節點分根節點、父節點和子節點的概念。
如圖：樹深=4; 5是根節點；同樣8與3的關係是父子節點關係。
二叉樹binary tree，則加了“二叉”（binary），意思是在樹中作區分。每個節點至多有兩個子（child）,left child & right child。二叉樹在很多例子中使用，比如二叉樹表示算術表示式。
如圖：1/8是左節點；2/3是右節點；
二、二叉搜尋樹 BST
顧名思義，二叉樹上又加了個搜尋的限制。其要求：每個節點比其左子樹元素大，比其右子樹元素小。
如圖：每個節點比它左子樹的任意節點大，而且比它右子樹的任意節點小
三、BST Java實現
直接上程式碼，對應程式碼分享在 Github 主頁
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. */ /** * 二叉搜尋樹(BST)實現 * * Created by bysocket on 16/7/7. */ public class BinarySearchTree { /** * 根節點 */ public static TreeNode root; public BinarySearchTree() { this.root = null; } /** * 查詢 * 樹深(N) O(lgN) * 1. 從root節點開始 * 2. 比當前節點值小,則找其左節點 * 3. 比當前節點值大,則找其右節點 * 4. 與當前節點值相等,查詢到返回TRUE * 5. 查詢完畢未找到, * @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. 從root節點開始 * 2. 如果root為空,root為插入值 * 迴圈: * 3. 如果當前節點值大於插入值,找左節點 * 4. 如果當前節點值小於插入值,找右節點 * @param key * @return */ public TreeNode insert (int key) { // 新增節點 TreeNode newNode = new TreeNode(key); // 當前節點 TreeNode current = root; // 上個節點 TreeNode parent = null; // 如果根節點為空 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; } } } } /** * 刪除節點 * 1.找到刪除節點 * 2.如果刪除節點左節點為空 , 右節點也為空; * 3.如果刪除節點只有一個子節點 右節點 或者 左節點 * 4.如果刪除節點左右子節點都不為空 * @param key * @return */ public TreeNode delete (int key) { TreeNode parent = root; TreeNode current = root; boolean isLeftChild = false; // 找到刪除節點 及 是否在左子樹 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; } } // 如果刪除節點左節點為空 , 右節點也為空 if (current.left == null && current.right == null) { if (current == root) { root = null; } // 在左子樹 if (isLeftChild == true) { parent.left = null; } else { parent.right = null; } } // 如果刪除節點只有一個子節點 右節點 或者 左節點 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; } } // 如果刪除節點左右子節點都不為空 else if (current.left != null && current.right != null) { // 找到刪除節點的後繼者 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; } /** * 獲取刪除節點的後繼者 * 刪除節點的後繼者是在其右節點樹種最小的節點 * @param deleteNode * @return */ public TreeNode getDeleteSuccessor(TreeNode deleteNode) { // 後繼者 TreeNode successor = null; TreeNode successorParent = null; TreeNode current = deleteNode.right; while (current != null) { successorParent = successor; successor = current; current = current.left; } // 檢查後繼者(不可能有左節點樹)是否有右節點樹 // 如果它有右節點樹,則替換後繼者位置,加到後繼者父親節點的左節點. 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); } } } /** * 節點 */ class TreeNode { /** * 節點值 */ int value; /** * 左節點 */ TreeNode left; /** * 右節點 */ TreeNode right; public TreeNode(int value) { this.value = value; left = null; right = null; } }
1. 節點資料結構
首先定義了節點的資料介面，節點分左節點和右節點及本身節點值。如圖
程式碼如下：
/** * 節點 */ class TreeNode { /** * 節點值 */ int value; /** * 左節點 */ TreeNode left; /** * 右節點 */ TreeNode right; public TreeNode(int value) { this.value = value; left = null; right = null; } }
2. 插入
插入，和刪除一樣會引起二叉搜尋樹的動態變化。插入相對刪處理邏輯相對簡單些。如圖插入的邏輯：
a. 從root節點開始
b.如果root為空,root為插入值
c.迴圈:
d.如果當前節點值大於插入值,找左節點
e.如果當前節點值小於插入值,找右節點
程式碼對應：
/**
* 插入
* 1. 從root節點開始
* 2. 如果root為空,root為插入值
* 迴圈:
* 3. 如果當前節點值大於插入值,找左節點
* 4. 如果當前節點值小於插入值,找右節點
* @param key
* @return
*/
public TreeNode insert (int key) {
// 新增節點
TreeNode newNode = new TreeNode(key);
// 當前節點
TreeNode current = root;
// 上個節點
TreeNode parent = null;
// 如果根節點為空
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.查詢
其演算法複雜度 : O(lgN),樹深(N)。如圖查詢邏輯：
a.從root節點開始
b.比當前節點值小,則找其左節點
c.比當前節點值大,則找其右節點
d.與當前節點值相等,查詢到返回TRUE
e.查詢完畢未找到
程式碼對應：
/**
* 查詢
* 樹深(N) O(lgN)
* 1. 從root節點開始
* 2. 比當前節點值小,則找其左節點
* 3. 比當前節點值大,則找其右節點
* 4. 與當前節點值相等,查詢到返回TRUE
* 5. 查詢完畢未找到,
* @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. 刪除
首先找到刪除節點，其尋找方法：刪除節點的後繼者是在其右節點樹種最小的節點。如圖刪除對應邏輯：
結果為：
a.找到刪除節點
b.如果刪除節點左節點為空 , 右節點也為空;
c.如果刪除節點只有一個子節點 右節點 或者 左節點
d.如果刪除節點左右子節點都不為空
程式碼對應見上面完整程式碼。
案例測試程式碼如下，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.
*/
/**
* 二叉搜尋樹(BST)測試案例 {@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);
// 列印二叉樹
b.toString(b.root);
System.out.println();
// 是否存在節點值10
TreeNode node01 = b.search(10);
System.out.println("是否存在節點值為10 => " + node01.value);
// 是否存在節點值11
TreeNode node02 = b.search(11);
System.out.println("是否存在節點值為11 => " + node02);
// 刪除節點8
TreeNode node03 = b.delete(8);
System.out.println("刪除節點8 => " + node03.value);
b.toString(b.root);
}
}
執行結果如下：
value = 1 -> value = 2 -> value = 3 -> value = 4 -> value = 6 -> value = 8 -> value = 9 -> value = 10 -> value = 20 -> value = 25 -> 是否存在節點值為10 => 10 是否存在節點值為11 => null 刪除節點8 => 8 value = 1 -> value = 2 -> value = 3 -> value = 4 -> value = 6 -> value = 9 -> value = 10 -> value = 20 -> value = 25 ->
四、小結
與偶爾吃一碗“老壇酸菜牛肉麵”一樣的味道，品味一個演算法，比如BST，的時候，總是那種說不出的味道。
樹，二叉樹的概念
BST演算法
相關程式碼分享在 Github 主頁