Binary search tree can be recursively defined as follows. A binary search tree is either an empty binary tree, or a binary tree that satisfies the following properties:
(1) If its left child If the tree is not empty, the value of the key of any node on its left subtree is smaller than the value of the key of the root node.
(2) If its right subtree is not empty, the value of the keyword of any node on its right subtree is greater than the value of the root node's keyword.
(3) Its left and right subtrees themselves are a binary search tree.
In terms of performance, if the number of nodes in the left and right subtrees of all non-leaf nodes of the binary search tree remains about the same (balanced), then the binary search tree The search performance is close to binary search; but its advantage over binary search in continuous memory space is that changing the binary search tree structure (inserting and deleting nodes) does not require moving large segments of memory data, often even with constant overhead. A binary search tree can represent a combination of data sets arranged in a sequential sequence, so a binary search tree is also called a binary sorting tree, and the same data set can be represented as different binary search trees. The data structure of the node of the binary search tree is defined as:
struct celltype{
records data;
celltype * lchild, * rchild;
}
typedef celltype * BST;In Java, the data structure of the node is defined as follows:
package wx.algorithm.search.bst;
/**
* Created by apple on 16/7/29.
*/
/**
* @function 二叉搜索树中的节点
*/
public class Node {
//存放节点数据
int data;
//指向左子节点
Node left;
//指向右子节点
Node right;
/**
* @function 默认构造函数
* @param data 节点数据
*/
public Node(int data) {
this.data = data;
left = null;
right = null;
}
}And the binary search tree The search process starts from the root node. If the keyword of query is equal to the keyword of the node, then it will hit; otherwise, if the query keyword is smaller than the node keyword, it will enter the left son. ; If it is larger than the node keyword, enter the right son; if the pointer of the left son or right son is empty, it reports that the corresponding keyword cannot be found.
BST Search(keytype k, BST F){
//在F所指的二叉查找树中查找关键字为k的记录。若成功，则返回响应结点的指针，否则返回空
if(F == NULL) //查找失败
return NULL;
else if(k == F -> data.key){ //查找成功
return F;
}
else if (k < F -> data.key){ //查找左子树
return Search(k,F -> lchild);
}
else if (k > F -> data.key){ //查找右子树
return Search(k,F -> rchild);
}
}Insert a new record R into the binary search tree. It should be ensured that the structural properties of the binary search tree are not destroyed after the insertion. . Therefore, in order to perform an insertion operation one should first look up where R is. When searching, the above-mentioned recursive algorithm is still used. If the search fails, the node containing R is inserted into the position of the empty subtree. If the search succeeds, the insertion is not performed and the operation ends.
void Insert(records R, BST &F){
//在F所指的二叉查找树中插入一个新纪录R
if(F == NULL){
F = new celltype;
F -> data = R;
F -> lchild = NULL;
F -> rchild = NULL;
}
else if (R.key < F -> data.key){
Insert(R,F -> lchild);
}else if(R.key > F -> data.key){
Insert(R,F -> rchild);
}
//如果 R.key == F -> data.key 则返回
}
If we perform a simple replacement, we may encounter the following situation:
So we need to choose a suitable replacement node in the subtree. Generally speaking, the replacement node will be the smallest node in the right subtree:
Java version code reference of BinarySearchTreeBinarySearchTree:
package wx.algorithm.search.bst;
/**
* Created by apple on 16/7/29.
*/
/**
* @function 二叉搜索树的示范代码
*/
public class BinarySearchTree {
//指向二叉搜索树的根节点
private Node root;
//默认构造函数
public BinarySearchTree() {
this.root = null;
}
/**
* @param id 待查找的值
* @return
* @function 默认搜索函数
*/
public boolean find(int id) {
//从根节点开始查询
Node current = root;
//当节点不为空
while (current != null) {
//是否已经查询到
if (current.data == id) {
return true;
} else if (current.data > id) {
//查询左子树
current = current.left;
} else {
//查询右子树
current = current.right;
}
}
return false;
}
/**
* @param id
* @function 插入某个节点
*/
public void insert(int id) {
//创建一个新的节点
Node newNode = new Node(id);
//判断根节点是否为空
if (root == null) {
root = newNode;
return;
}
//设置current指针指向当前根节点
Node current = root;
//设置父节点为空
Node parent = null;
//遍历直到找到第一个插入点
while (true) {
//先将父节点设置为当前节点
parent = current;
//如果小于当前节点的值
if (id < current.data) {
//移向左节点
current = current.left;
//如果当前节点不为空,则继续向下一层搜索
if (current == null) {
parent.left = newNode;
return;
}
} else {
//否则移向右节点
current = current.right;
//如果当前节点不为空,则继续向下一层搜索
if (current == null) {
parent.right = newNode;
return;
}
}
}
}
/**
* @param id
* @return
* @function 删除树中的某个元素
*/
public boolean delete(int id) {
Node parent = root;
Node current = root;
//记录被找到的节点是父节点的左子节点还是右子节点
boolean isLeftChild = false;
//循环直到找到目标节点的位置,否则报错
while (current.data != id) {
parent = current;
if (current.data > id) {
isLeftChild = true;
current = current.left;
} else {
isLeftChild = false;
current = current.right;
}
if (current == null) {
return false;
}
}
//如果待删除的节点没有任何子节点
//直接将该节点的原本指向该节点的指针设置为null
if (current.left == null && current.right == null) {
if (current == root) {
root = null;
}
if (isLeftChild == true) {
parent.left = null;
} else {
parent.right = null;
}
}
//如果待删除的节点有一个子节点,且其为左子节点
else if (current.right == null) {
//判断当前节点是否为根节点
if (current == root) {
root = current.left;
} else if (isLeftChild) {
//挂载到父节点的左子树
parent.left = current.left;
} else {
//挂载到父节点的右子树
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;
}
}
//如果待删除的节点有两个子节点
else if (current.left != null && current.right != null) {
//寻找右子树中的最小值
Node successor = getSuccessor(current);
if (current == root) {
root = successor;
} else if (isLeftChild) {
parent.left = successor;
} else {
parent.right = successor;
}
successor.left = current.left;
}
return true;
}
/**
* @param deleleNode
* @return
* @function 在树种查找最合适的节点
*/
private Node getSuccessor(Node deleleNode) {
Node successsor = null;
Node successsorParent = null;
Node current = deleleNode.right;
while (current != null) {
successsorParent = successsor;
successsor = current;
current = current.left;
}
if (successsor != deleleNode.right) {
successsorParent.left = successsor.right;
successsor.right = deleleNode.right;
}
return successsor;
}
/**
* @function 以中根顺序遍历树
*/
public void display() {
display(root);
}
private void display(Node node) {
//判断当前节点是否为空
if (node != null) {
//首先展示左子树
display(node.left);
//然后展示当前根节点的值
System.out.print(" " + node.data);
//最后展示右子树的值
display(node.right);
}
}
}Test function:
package wx.algorithm.search.bst;
import org.junit.Before;
import org.junit.Test;
/**
* Created by apple on 16/7/30.
*/
public class BinarySearchTreeTest {
BinarySearchTree binarySearchTree;
@Before
public void setUp() {
binarySearchTree = new BinarySearchTree();
binarySearchTree.insert(3);
binarySearchTree.insert(8);
binarySearchTree.insert(1);
binarySearchTree.insert(4);
binarySearchTree.insert(6);
binarySearchTree.insert(2);
binarySearchTree.insert(10);
binarySearchTree.insert(9);
binarySearchTree.insert(20);
binarySearchTree.insert(25);
binarySearchTree.insert(15);
binarySearchTree.insert(16);
System.out.println("原始的树 : ");
binarySearchTree.display();
System.out.println("");
}
@Test
public void testFind() {
System.out.println("判断4是否存在树中 : " + binarySearchTree.find(4));
}
@Test
public void testInsert() {
}
@Test
public void testDelete() {
System.out.println("删除值为2的节点 : " + binarySearchTree.delete(2));
binarySearchTree.display();
System.out.println("\n 删除有一个子节点值为4的节点 : " + binarySearchTree.delete(4));
binarySearchTree.display();
System.out.println("\n 删除有两个子节点的值为10的节点 : " + binarySearchTree.delete(10));
binarySearchTree.display();
}
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