【数据结构】二叉搜索树

二叉搜索树，又叫二叉排序树，二叉查找树。它有以下特点：

左子树的值小于根节点的值，右子树的值大于根节点的值；二叉搜索树中序遍历的结果

就是一个升序序列。

当然，空树也是一个二叉搜索树。

全局满足二叉搜索树的性质，局部也应该满足。

既然有以上性质，那么二叉树的查找是相当方便的，当然插入和删除，复杂度也会明显

降低。

查找：从根节点开始，如果要插入的key小于根节点的key，则向左走，否则向右走，找

到了直接返回，如果已经走到空了，还没找到，说明要查找的key不存在。下边给出实现

代码：

bool FindNonR(const K& key){if (NULL == _root)return false;Node* cur = _root;while (cur){if (cur->_key < key){cur = cur->_right;}else if (cur->_key > key){cur = cur->_left;}elsereturn true;}return false;}

递归版本比较简单，这里就不给出实现了。

插入：找到合适的空位置，插入元素，如果要插入的值，在原来的搜索二叉树中已将存

在则不能进行插入。下边画图说明：

根据上边的图解，写出如下代码：

bool InsertNonR(const K& key){if (NULL == _root)//空树{_root = new Node(key);return true;}//不是空树Node* cur = _root;Node* parent = NULL;while (cur){if (cur->_key < key){parent = cur;cur = cur->_right;}else if (cur->_key > key){parent = cur;cur = cur->_left;}else//已经存在的值{return false;}}Node* NewNode = new Node(key);if (key < parent->_key) parent->_left = NewNode ;elseparent->_right = NewNode;return true;}

从上边的图中，我们可以看出，插入元素必须记住父节点，否则无法实现连接操作。

二叉搜索树也可以递归实现：下边给出实现代码：

bool _InsertR(Node*& root, const K& key){if (root == NULL){root = new Node(key);return true;}Node* cur = root;if (cur->_key < key){_InsertR(cur->_right,key);}else if (cur->_key > key){_InsertR(cur->_left, key);}elsereturn false; }

这段代码中，我们可以看出：形参中我们采用的是引用传参，这样巧妙的利用了引用的

性质：变量的别名，两者同时被修改。

当原来的树就是空树（_root == NULL），root的改变就直接影响了_root的改变；

当原来的树中已经有一些元素,（利用上图的例子，插入3）

（前边一些查找步骤省略）4的left是空，走到函数第一个if，创建一个新的结点，连在

root上，此时的root是4的left的别名，所以两者都改变了，起到了连接的作用。

删除：找到要删除的元素，进行删除，这个比较复杂，图解：

下边给出代码实现：

bool RemoveNonR(const K& key){if (NULL == _root)return false;Node* cur = _root;Node* parent = NULL;while (cur){if (cur->_key < key){parent = cur;cur = cur->_right;}else if (cur->_key > key){parent = cur;cur = cur->_left;}else{Node* del = NULL;//叶子节点直接删除if (cur->_left == NULL && cur->_right == NULL){if (parent && cur == parent->_left){parent->_left = NULL;del = cur;}else if (parent && cur == parent->_right){parent->_right = NULL;del = cur;}else{del = cur;_root = NULL;}}//只有一个孩子的结点else if (cur->_left == NULL || cur->_right == NULL){//只有左孩子if (cur->_left != NULL){if (parent == NULL){_root = parent->_left;del = cur;}else if (parent->_left == cur){parent->_left = cur->_left;}else{parent->_right = cur->_left;}}//只有右孩子else{if (parent == NULL){_root = cur->_right;del = cur;}else if (parent->_right == cur){parent->_right = cur->_right;}else//cur是parent的左孩子{parent->_left = cur->_right;}}}else//有两个孩子的结点{Node* tmp = cur->_right;//查找右子树的最左结点while (tmp && tmp->_left){parent = tmp;tmp = tmp->_left;}swap(cur->_key,tmp->_key);//删除tmp结点//tmp是叶子节点 ，tmp只有一个孩子结点if (tmp == parent->_left){parent->_left = tmp->_right;}else{parent->_right = tmp->_right;}del = tmp;}delete del;del = NULL;return true;}}//没有找到要删除的值return false;}

对于删除操作，也可以用递归实现：

代码：

bool _RemoveR(Node*& root,const K& key){if (root == NULL)return false;if (root->_key < key){_RemoveR(root->_right,key);}else if (root->_key > key){_RemoveR(root->_left,key);}else{Node* del = root;Node* parent = root;if (root->_left == NULL)//仅有右孩子{root = root->_right;}else if (root->_right ==NULL){root = root->_left;}else{Node* minRight = root->_right;//找右子树的最左结点while (minRight->_left){parent = minRight;minRight = minRight->_left;}root->_key = minRight->_key;if (minRight == parent->_right){parent->_right = minRight->_right;}else{parent->_left = minRight->_right;}del = minRight;}delete del;}return true;}

这段代码中，找要删除的结点是递归操作。找到之后，如果该节点只有一个孩子或者没

有孩子，直接将指向当前结点的指针指向其孩子结点或者空，利用引用，很好的连接了

起来。如果当前结点有左右孩子，利用非递归中的处理办法解决。

【完整代码】

#pragma once#include<iostream>using namespace std;#include<stack>template<typename K>struct SearchBinaryTreeNode{K _key;SearchBinaryTreeNode<K>* _left;SearchBinaryTreeNode<K>* _right;SearchBinaryTreeNode(const K& key):_key(key),_left(NULL),_right(NULL){}};template<typename K>class SearchBinaryTree{typedef SearchBinaryTreeNode<K> Node;public:SearchBinaryTree():_root(NULL){}void InorderNonR(){_InorderNonR(_root);}void InsertR(const  K& key){_InsertR(_root,key);}void InOrderR(){_InOrderR(_root);cout << endl;}void RemoveR(const K& key){_RemoveR(_root,key);}bool RemoveNonR(const K& key){if (NULL == _root)return false;Node* cur = _root;Node* parent = NULL;while (cur){if (cur->_key < key){parent = cur;cur = cur->_right;}else if (cur->_key > key){parent = cur;cur = cur->_left;}else{Node* del = NULL;//叶子节点直接删除if (cur->_left == NULL && cur->_right == NULL){if (parent && cur == parent->_left){parent->_left = NULL;del = cur;}else if (parent && cur == parent->_right){parent->_right = NULL;del = cur;}else{del = cur;_root = NULL;}}//只有一个孩子的结点else if (cur->_left == NULL || cur->_right == NULL){//只有左孩子if (cur->_left != NULL){if (parent == NULL){_root = parent->_left;del = cur;}else if (parent->_left == cur){parent->_left = cur->_left;}else{parent->_right = cur->_left;}}//只有右孩子else{if (parent == NULL){_root = cur->_right;del = cur;}else if (parent->_right == cur){parent->_right = cur->_right;}else//cur是parent的左孩子{parent->_left = cur->_right;}}}else//有两个孩子的结点{Node* tmp = cur->_right;//查找右子树的最左结点while (tmp && tmp->_left){parent = tmp;tmp = tmp->_left;}swap(cur->_key,tmp->_key);//删除tmp结点//tmp是叶子节点 ，tmp只有一个孩子结点if (tmp == parent->_left){parent->_left = tmp->_right;}else{parent->_right = tmp->_right;}del = tmp;}delete del;del = NULL;return true;}}//没有找到要删除的值return false;}bool FindNonR(const K& key){if (NULL == _root)return false;Node* cur = _root;while (cur){if (cur->_key < key){cur = cur->_right;}else if (cur->_key > key){cur = cur->_left;}elsereturn true;}return false;}bool InsertNonR(const K& key){if (NULL == _root)//空树{_root = new Node(key);return true;}//不是空树Node* cur = _root;Node* parent = NULL;while (cur){if (cur->_key < key){parent = cur;cur = cur->_right;}else if (cur->_key > key){parent = cur;cur = cur->_left;}else//已经存在的值{return false;}}Node* NewNode = new Node(key);if (key < parent->_key) parent->_left = NewNode ;elseparent->_right = NewNode;return true;}protected:bool _RemoveR(Node*& root,const K& key){if (root == NULL)return false;if (root->_key < key){_RemoveR(root->_right,key);}else if (root->_key > key){_RemoveR(root->_left,key);}else{Node* del = root;Node* parent = root;if (root->_left == NULL)//仅有右孩子{root = root->_right;}else if (root->_right ==NULL){root = root->_left;}else{Node* minRight = root->_right;//找右子树的最左结点while (minRight->_left){parent = minRight;minRight = minRight->_left;}root->_key = minRight->_key;if (minRight == parent->_right){parent->_right = minRight->_right;}else{parent->_left = minRight->_right;}del = minRight;}delete del;}return true;}void _InOrderR(Node* root){if (root == NULL)return;_InOrderR(root->_left);cout << root->_key << " ";_InOrderR(root->_right);}bool _InsertR(Node*& root, const K& key){if (root == NULL){root = new Node(key);return true;}Node* cur = root;if (cur->_key < key){_InsertR(cur->_right,key);}else if (cur->_key > key){_InsertR(cur->_left, key);}elsereturn false; }void _InorderNonR(Node* root){if (NULL != root){Node* cur = root;stack<Node*> s;while (cur || !s.empty()){while (cur){s.push(cur);cur = cur->_left;}Node* top = s.top();s.pop();cout << top->_key << " ";cur = top->_right;}}cout << endl;}private:Node* _root;};void TestBinaySearchNonR(){SearchBinaryTree<int> bs;//int array[] = { 2,4,1,3,6,5 };int array[] = {5,3,4,1,7,8,2,6,0,9};for (size_t i = 0; i < 10; ++i){bs.InsertNonR(array[i]);}bs.InorderNonR();bs.FindNonR(3);bs.RemoveNonR(0);bs.InorderNonR();bs.RemoveNonR(1);bs.RemoveNonR(2);bs.RemoveNonR(3);bs.RemoveNonR(4);bs.RemoveNonR(5);bs.RemoveNonR(6);bs.RemoveNonR(7);bs.RemoveNonR(8);bs.RemoveNonR(9);bs.InorderNonR();}void TestBinaySearchR(){SearchBinaryTree<int> bs;int array[] = { 5,3,4,1,7,8,2,6,0,9 };for (size_t i = 0; i < 10; ++i){bs.InsertR(array[i]);}bs.InOrderR();cout <<bs.FindNonR(3)<<endl;bs.RemoveR(0);bs.InorderNonR();bs.RemoveR(1);bs.RemoveR(2);bs.RemoveR(3);bs.InorderNonR();bs.RemoveR(4);bs.RemoveR(5);bs.RemoveR(6);cout << bs.FindNonR(3) << endl;bs.RemoveR(7);bs.RemoveR(8);bs.RemoveR(9);bs.InOrderR();}