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1、二叉搜索树的概念

二叉搜索树又叫做二叉排序树，他或者是一棵空树，或者具有以下性质：

若他的左子树不为空，则左子树的所有节点的值都小于根节点的值，

若他的右子树不为空，则右子树的所有节点的值都大于根节点的值，

他的左右子树也分别为二叉搜索树；

2、二叉搜索树的操作

  int a[] = {8, 3, 1, 10, 6, 4, 7, 14, 13};

1.二叉搜索树的查找

从根节点开始找，比根大往右边查找，比根小往左边查找，最多查找高度次，走到空还没找到，则该值不存在；

2.二叉搜索树的插入

若树为空，则新增节点，赋值给root指针，

若树不为空，按二叉搜索树查找插入位置，插入新节点

 3.二叉搜索树的删除

首先查找元素是否在二叉搜索树中，如果不存在，则返回，如果存在，则分为以下四种情况：

a. 要删除的结点无孩子结点

b. 要删除的结点只有左孩子结点

c. 要删除的结点只有右孩子结点

d. 要删除的结点有左、右孩子结点

4.二叉搜索树的实现

#pragma once#include<iostream>
#include<string>
using namespace std;
namespace key
{template<class K>struct BSTreeNode{BSTreeNode<K>* _left;BSTreeNode<K>* _right;K _key;BSTreeNode(const K& key):_left(nullptr),_right(nullptr),_key(key){ }};template<class K>class BSTree{typedef BSTreeNode<K> Node;public:bool Insert(const K& key){if (_root == nullptr){_root = new Node(key);return true;}Node* parent = nullptr;Node* cur = _root;while (cur){if (cur->_key < key){parent = cur;cur = cur->_right;}else if (cur->_key > key){parent = cur;cur = cur->_left;}else{return false;}}cur = new Node(key);if (parent->_key < key){parent->_right = cur;}else{parent->_left = cur;}return true;}bool Find(const K& key){Node* cur = _root;while (cur){if (cur->_key < key){cur = cur->_right;}else if(cur->_key>key){cur = cur->_left;}else{cout << "true" << endl;return true;}}cout << "false" << endl;return false;}bool Erase(const K& key){Node* parent = nullptr;Node* cur = _root;while (cur){if (cur->_key < key){parent = cur;cur = cur->_right;}else if (cur->_key > key){parent = cur;cur = cur->_left;}else{//删除//左为空，父亲指向我的右if (cur->_left == nullptr){if (cur == _root){_root = cur->_right;}else{if (parent->_left == cur){parent->_left = cur->_right;}else{parent->_right=cur->_right;}}delete cur;}//右为空，父亲指向我的左else if (cur->_right == nullptr){if (cur == _root){_root = cur->_left;}else{if (parent->_left == cur){parent->_left = cur->_left;}else{parent->_right = cur->_left;}}delete cur;}else{//左右都不为空，替换法//查找右子树的最小节点或左子树的最大节点//我们这里找右子树的最小节点(也就是最左节点)Node* rightMinParent = cur;Node* rightMin = cur->_right;while (rightMin->_left){rightMinParent = rightMin;rightMin = rightMin->_left;}swap(cur->_key, rightMin->_key);if (rightMinParent->_left == rightMin){rightMinParent->_left = rightMin->_right;}else{rightMinParent->_right = rightMin->_right;}delete rightMin;}return true;}}return false;}void InOrder(){_InOrder(_root);cout << endl;}private:void _InOrder(Node* root){if (root == nullptr){return;}_InOrder(root->_left);cout << root->_key << " ";_InOrder(root->_right);}Node* _root = nullptr;};void TestBSTree1(){BSTree<int> b;b.Insert(1);b.Insert(2);b.Insert(3);b.Insert(4);b.Insert(5);b.Find(6);b.Find(3);b.Find(5);b.InOrder();}void TestBSTree2(){int a[] = { 8, 3, 1, 10, 6, 4, 7, 14, 13 };BSTree<int> t1;for (auto e : a){t1.Insert(e);}/*t1.InOrder();t1.Erase(8);t1.InOrder();*/for (auto e : a){t1.Erase(e);t1.InOrder();}}
}namespace key_value
{template<class K,class V>struct BSTreeNode{BSTreeNode<K,V>* _left;BSTreeNode<K,V>* _right;K _key;V _value;BSTreeNode(const K& key,const V& value):_left(nullptr), _right(nullptr), _key(key), _value(value){ }};template<class K,class V>class BSTree{typedef BSTreeNode<K,V> Node;public:bool Insert(const K& key, const V& value){if (_root == nullptr){_root = new Node(key,value);return true;}Node* parent = nullptr;Node* cur = _root;while (cur){if (cur->_key < key){parent = cur;cur = cur->_right;}else if (cur->_key > key){parent = cur;cur = cur->_left;}else{return false;}}cur = new Node(key,value);if (parent->_key < key){parent->_right = cur;}else{parent->_left = cur;}return true;}Node* Find(const K& key){Node* cur = _root;while (cur){if (cur->_key < key){cur = cur->_right;}else if (cur->_key > key){cur = cur->_left;}else{return cur;}}return cur;}bool Erase(const K& key){Node* parent = nullptr;Node* cur = _root;while (cur){if (cur->_key < key){parent = cur;cur = cur->_right;}else if (cur->_key > key){parent = cur;cur = cur->_left;}else{//删除//左为空，父亲指向我的右if (cur->_left == nullptr){if (cur == _root){_root = cur->_right;}else{if (parent->_left == cur){parent->_left = cur->_right;}else{parent->_right = cur->_right;}}delete cur;}//右为空，父亲指向我的左else if (cur->_right == nullptr){if (cur == _root){_root = cur->_left;}else{if (parent->_left == cur){parent->_left = cur->_left;}else{parent->_right = cur->_left;}}delete cur;}else{//左右都不为空，替换法//查找右子树的最小节点或左子树的最大节点//我们这里找右子树的最小节点(也就是最左节点)Node* rightMinParent = cur;Node* rightMin = cur->_right;while (rightMin->_left){rightMinParent = rightMin;rightMin = rightMin->_left;}swap(cur->_key, rightMin->_key);if (rightMinParent->_left == rightMin){rightMinParent->_left = rightMin->_right;}else{rightMinParent->_right = rightMin->_right;}delete rightMin;}return true;}}return false;}void InOrder(){_InOrder(_root);cout << endl;}private:void _InOrder(Node* root){if (root == nullptr){return;}_InOrder(root->_left);cout << root->_key << ":" << root->_value << " ";_InOrder(root->_right);}Node* _root = nullptr;};void TestBSTree3(){BSTree<string, string> dict;dict.Insert("string", "字符串");dict.Insert("left", "左边");dict.Insert("insert", "插入");string str;while (cin >> str){BSTreeNode<string, string>* ret = dict.Find(str);if (ret){cout << ret->_value << endl;}else{cout << "无此单词，请重新输入" << endl;}}}void TestBSTree4(){// 统计次数string arr[] = { "苹果", "西瓜", "苹果", "西瓜", "苹果", "苹果", "西瓜","苹果", "香蕉", "苹果", "香蕉","苹果","草莓", "苹果","草莓" };BSTree<string, int> countTree;for (const auto& str : arr){auto ret = countTree.Find(str);if (ret == nullptr){countTree.Insert(str, 1);}else{ret->_value++;}}countTree.InOrder();}}

5、二叉搜索树的应用

1.K模型：K模型只有key作为关键码，结构中只需要存储key即可，关键码即为需要搜索到的值；

比如：给一个单词word，判断该单词是否拼写正确，

2.K-V模型：每一个关键码都对应一个Value，即<Key,value>的键值对，

比如：英汉字典中用英文与中文的对应关系，通过英文可以快速找到对应的中文，

6、二叉搜索树的性能分析

对于有n个节点的二叉搜索树，若每个元素查找的概率相等，则二叉搜索树平均查找长度是高度次，但对于同一个关键码集合，如果插入的次序不同，可能得到不同结构的二叉树：