【数据结构】：二叉搜索树

二叉搜索树，也称有序二叉树,排序二叉树，是指一棵空树或者具有下列性质的二叉树：

1. 若任意节点的左子树不空，则左子树上所有结点的值均小于它的根结点的值；

2. 若任意节点的右子树不空，则右子树上所有结点的值均大于它的根结点的值；

3. 任意节点的左、右子树也分别为二叉查找树。

4. 没有键值相等的节点。

因此，搜索二叉树中没有键值冗余的节点，通常可用来去重排序。

如图所示就是一棵二叉搜索树：

中序遍历结果为：0 1 2 3 4 5 6 7 8 9

需要注意的是：二叉搜索树在进行删除和插入操作后都需要再次进行调整，新得到的树也应该是二叉搜索树。

具体代码实现如下：

#include<iostream>
using namespace std;

template<class K>
class BinaraySearchNode
{
public:
    BinaraySearchNode<K>* _left;
    BinaraySearchNode<K>* _right;
    BinaraySearchNode<K>* _parent;
    K  _key;

    BinaraySearchNode(const K& key)
        :_left(NULL),_right(NULL),_parent(NULL),_key(key)
    {}
};

template<class K>
class BinarySearch
{
public:
    typedef BinaraySearchNode<K> Node;

    BinarySearch()
        :_root(NULL)
    {}

    //插入
    bool InSert(const K& key)
    {
        if (_root == NULL)
        {
            _root = new Node(key);
            return true;
        }
        Node* parent = NULL;
        Node* cur = _root;
        while (cur)
        {
            if (cur->_key > key)
            {
                parent = cur;
                cur = cur->_left;
            }
            else if (cur->_key < key)
            {
                parent = cur;
                cur = cur->_right;
            }
            else
                return false;
        }
        Node* tmp = new Node(key);
        if (parent->_key > key)
            parent->_left = tmp;
        else if (parent->_key < key)
            parent->_right = tmp;
        return true;
    }

    //查找
    Node* Find(const K& key)const
    {
        Node* cur = _root;
        while (cur)
        {
            if (cur->_key > key)
                cur = cur->_left;
            else if (cur->_key < key)
                cur = cur->_right;
            return cur;
        }
        return NULL;
    }

    //删除节点
    bool Remove(const K& key)
    {
        if (_root == NULL)
            return false;
        Node* parent = NULL;
        Node* cur = _root;
        while (cur)
        {
            if (key > cur->_key)
            {
                parent = cur;
                cur = cur->_right;
            }
            else if (key < cur->_key)
            {
                parent = cur;
                cur = cur->_left;
            }
            else//等于
            {
                if (cur->_left == NULL)
                {
                    parent = cur->_parent;
                    if (parent == NULL)
                    {
                        _root = cur->_right;
                        delete cur;
                        cur = NULL;
                        return true;
                    }
                    if (cur == parent->_left)
                    {
                        parent->_left = cur->_right;
                    }
                    else
                    {
                        parent->_right = cur->_right;
                    }
                    delete cur;
                    cur = NULL;
                }
                if (cur->_right == NULL)
                {
                    parent = cur->_parent;
                    if (parent == NULL)
                    {
                        _root = cur->_left;
                        delete cur;
                        cur = NULL;
                        return true;
                    }
                    if (cur == parent->_left)
                    {
                        parent->_left = cur->_left;
                    }
                    else
                    {
                        parent->_right = cur->_left;
                    }
                    delete cur;
                    cur = NULL;
                }
                if (cur->_left != NULL && cur->_right != NULL)
                {
                    //替换法(注意根节点为空的情况）
                    Node* subRight = cur->_right;
                    Node* subParent = cur;
                    while (subRight->_left)
                    {
                        subParent = subRight;
                        subRight = subRight->_left;
                    }

                    cur->_key = subRight->_key;
                    if (subParent->_right == subRight)
                        subParent->_right = subRight->_right;
                    else
                        subParent->_left = subRight->_left;
                    delete subRight;
                    subRight = NULL;
                }
                return  true;
            }
        }
        return false;
    }

    //递归插入
    bool _InSertR(Node*& root,const K& key)//递归插入
    {
        if (root == NULL)
            return root = new Node(key);
        if (root->_key < key)
            return _InSertR(root->right,key);
        if (root->_key > key)
            return _InSertR(root->_left,key);
    }


    //递归删除
    bool _RemoveR(Node*& root,const K& key)//注意：这里要加引用
    {
        if (root == NULL)
            return false;
        if (root->_left == NULL && root->_right == NULL)
        {
            if (root->_key == key)
            {
                delete root;
                return true;
            }
            else
                return false;
        }
        if (root->_key > key)
            RemoveR(root->_left,key);
        if (root->_key < key)
            RemoveR(root->_right,key);
        else
        {
            Node* cur = root;
            if (cur->_left == NULL)
            {
                parent = cur->_parent;
                if (parent == NULL)
                {
                    root = root->_right;
                    delete cur;
                    cur = NULL;
                }
                if (cur == parent->_left)
                {
                    parent->_left = cur->_right;
                }
                else
                {
                    parent->_right = cur->_right;
                }
                delete cur;
                cur = NULL;
            }
            if (cur->_right == NULL)
            {
                parent = cur->_parent;
                if (parent == NULL)
                {
                    _root = cur->_left;
                    delete cur;
                    cur = NULL;
                    return true;
                }
                if (cur == parent->_left)
                {
                    parent->_left = cur->_left;
                }
                else
                {
                    parent->_right = cur->_left;
                }
                delete cur;
                cur = NULL;
            }
            if (cur->_left != NULL && cur->_right != NULL)
            {
                //替换法
                Node* subRight = cur->_right;
                Node* subParent = cur;
                while (subRight->_left)
                {
                    subParent = subRight;
                    subRight = subRight->_left;
                }

                cur->_key = subRight->_key;
                if (subParent->_right == subRight)
                    subParent->_right = subRight->_right;
                else
                    subParent->_left = subRight->_left;
                delete subRight;
                subRight = NULL;
            }
            return  true;
        }
    }

    //中序遍历
    void Inorder()
    {
        _Inorder(_root);
    }
protected:
    void _Inorder(Node* root)
    {
        if (root == NULL)
            return;
        else
        {
            _Inorder(root->_left);
            cout << root->_key << " ";
            _Inorder(root->_right);
        }
    }
private:
    Node* _root;
};

void test()
{
    BinarySearch<int> bs;
    int a[] = { 5,3,1,4,7,6,8,9,2}; for (size_t i = 0; i < (sizeof(a) / sizeof(a[0])); ++i) { bs.InSert(a[i]); } bs.Inorder(); cout << endl; bs.Remove(5); bs.Inorder(); }; 

#define _CRT_SECURE_NO_WARNINGS 1
#include"binarysearch.h"


int main()
{
    test();
    system("pause");
    return 0;
}