二叉搜索树的定义、查找、插入和删除

二叉搜索树的定义

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。





二叉搜索树的删除：



具体实现过程解析：

二叉搜索树的结构实现：

//二叉搜索树结构
template<class K, class V>
struct BSTreeNode
{
BSTreeNode* _left;
BSTreeNode* _right;
K _key;
V _value;

BSTreeNode(const K& key, const V& value)
:_left(NULL)
,_right(NULL)
,_key(key)
,_value(value)
{}

};

查找实现有迭代和递归两种

迭代法：

//在二叉搜索树中查找节点
Node* Find(const K& key)
{
Node* cur=_root;
//开始遍历查找
while (cur)
{
if (cur->_key > key)
{
cur = cur->_left;
}
else if(cur->_key<key)
{
cur = cur->_right;
}
else
{
return cur;
}
}

return NULL;
} 递归法：

//递归查找法
Node* _Find_R(Node* root, const K& key)
{
if (root == NULL)
{
return NULL;
}
if (root->_key > key)
{
return _Find_R(root->_left, key);
}
else if (root->_key < key)
{
return _Find_R(root->_right, key);
}
else
{
return root;
}
}

删除迭代法：

//在二叉搜索树中删除节点
bool Remove(const K& key)
{
//没有节点
if (_root == NULL)
{
return false;
}
//只有一个节点
if (_root->_left == NULL&&_root->_right == NULL)
{
if (_root->_key == key)
{
delete _root;
_root = NULL;
return true;
}

return false;
}

Node* parent = NULL;
Node* cur = _root;
//遍历查找要删除节点的位置
while (cur)
{
Node* del = NULL;
if (cur->_key > key)
{
parent = cur;
cur = cur->_left;
}
else if (cur->_key < key)
{
parent = cur;
cur = cur->_right;
}
else
{
//要删除节点的左子树为空，分3种情况
if (cur->_left == NULL)
{
//注意判断父节点是否为空，若为空，则要删除的节点为根节点,如：只有根节点5和其右节点9
if (parent == NULL)
{
_root = cur->_right;
delete cur;
cur = NULL;
return true;
}
if (parent->_key > cur->_key)
{
del = cur;
parent->_left = cur->_right;
delete del;
return true;
}
else if (parent->_key < key)
{
del = cur;
parent->_right = cur->_right;
delete del;
return true;
}
}
//要删除节点的右子树为空，同样分3种情况
else if (cur->_right == NULL)
{
//注意判断父节点是否为空，若为空，则要删除的节点为根节点,如：只有根节点5和其左节点3
if (parent == NULL)
{
_root = cur->_left;
delete cur;
cur = NULL;
return true;
}
if (parent->_key > cur->_key)
{
del = cur;
parent->_left = cur->_left;
delete del;
return true;
}
else if (parent->_key < cur->_key)
{
del = cur;
parent->_right = cur->_left;
delete del;
return true;
}
}
//左右子树都不为空
else
{
Node* del = cur;
Node* parent = NULL;
Node* RightFirst = cur->_right;
//右边第一个节点的左子树为空
if (RightFirst->_left == NULL)
{
swap(RightFirst->_key, cur->_key);
swap(RightFirst->_value, cur->_value);
del = RightFirst;
cur->_right = RightFirst->_right;
delete del;
return true;
}
//右边第一个节点的左子树不为空
while (RightFirst->_left)
{
parent = RightFirst;
RightFirst = RightFirst->_left;
}
swap(RightFirst->_key, cur->_key);
swap(RightFirst->_value, cur->_value);
del = RightFirst;
parent->_left = RightFirst->_right;
delete del;
return true;
}
}
}
return false;
}

删除递归法：

bool _Remove_R(Node*& root, const K& key)
{
//没有节点
if (root == NULL)
{
return false;
}
//只有一个节点
if (root->_left == NULL&&root->_right == NULL)
{
if (root->_key == key)
{
delete root;
root = NULL;
return true;
}
else
{
return false;
}

}

//删除二叉搜索树节点的递归写法
if (root->_key > key)
{
_Remove_R(root->_left, key);
}
else if (root->_key < key)
{
_Remove_R(root->_right, key);
}
else
{
Node* del = NULL;

if (root->_left == NULL)
{
del = root;
root = root->_right;
delete del;
del = NULL;
return true;
}
else if (root->_right == NULL)
{
del = root;
root = root->_left;
delete del;
del = NULL;
return true;
}
else
{
Node* RightFirst = root->_right;

while (RightFirst->_left)
{
RightFirst = RightFirst->_left;
}

swap(root->_key, RightFirst->_key);
swap(root->_value, RightFirst->_value);

_Remove_R(root->_right, key);
return true;
}
}
} 插入非递归：

//在二叉搜索树中插入节点
bool Insert(const K& key, const V& value)
{
if (_root == NULL)
{
_root = new Node(key, value);
}

Node* cur=_root;
Node* parent = NULL;
//首先找到要插入的位置
while (cur)
{
if (cur->_key > key)
{
parent = cur;
cur = cur->_left;
}
else if(cur->_key<key)
{
parent = cur;
cur = cur->_right;
}
else
{
return false;
}
}
//在找到插入位置以后，判断插入父亲节点的左边还是右边
if (parent->_key > key)
{
parent->_left = new Node(key, value);
}
else
{
parent->_right = new Node(key, value);
}

return true;
} 插入递归：

//递归插入法
bool _Insert_R(Node*& root, const K& key, const V& value)
{
if (root == NULL)
{
root = new Node(key, value);
return true;
}
if (root->_key > key)
{
return _Insert_R(root->_left, key, value);
}
else if(root->_key < key)
{
return _Insert_R(root->_right, key, value);
}
else
{
return false;
}
}


当二叉搜索树出现如下图情形时，效率最低：



完整代码及测试实现如下:

#include<iostream>
using namespace std;

//二叉搜索树结构
template<class K, class V>
struct BSTreeNode
{
BSTreeNode* _left;
BSTreeNode* _right;
K _key;
V _value;

BSTreeNode(const K& key, const V& value)
:_left(NULL)
,_right(NULL)
,_key(key)
,_value(value)
{}

};

template<class K,class V>
class BSTree
{
typedef BSTreeNode<K, V> Node;
public:
BSTree()
:_root(NULL)
{}

//在二叉搜索树中插入节点
bool Insert(const K& key, const V& value)
{
if (_root == NULL)
{
_root = new Node(key, value);
}

Node* cur=_root;
Node* parent = NULL;
//首先找到要插入的位置
while (cur)
{
if (cur->_key > key)
{
parent = cur;
cur = cur->_left;
}
else if(cur->_key<key)
{
parent = cur;
cur = cur->_right;
}
else
{
return false;
}
}
//在找到插入位置以后，判断插入父亲节点的左边还是右边
if (parent-&
d337
gt;_key > key)
{
parent->_left = new Node(key, value);
}
else
{
parent->_right = new Node(key, value);
}

return true;
}

//在二叉搜索树中查找节点
Node* Find(const K& key)
{
Node* cur=_root;
//开始遍历查找
while (cur)
{
if (cur->_key > key)
{
cur = cur->_left;
}
else if(cur->_key<key)
{
cur = cur->_right;
}
else
{
return cur;
}
}

return NULL;
}

//在二叉搜索树中删除节点
bool Remove(const K& key)
{
//没有节点
if (_root == NULL)
{
return false;
}
//只有一个节点
if (_root->_left == NULL&&_root->_right == NULL)
{
if (_root->_key == key)
{
delete _root;
_root = NULL;
return true;
}

return false;
}

Node* parent = NULL;
Node* cur = _root;
//遍历查找要删除节点的位置
while (cur)
{
Node* del = NULL;
if (cur->_key > key)
{
parent = cur;
cur = cur->_left;
}
else if (cur->_key < key)
{
parent = cur;
cur = cur->_right;
}
else
{
//要删除节点的左子树为空，分3种情况
if (cur->_left == NULL)
{
//注意判断父节点是否为空，若为空，则要删除的节点为根节点,如：只有根节点5和其右节点9
if (parent == NULL)
{
_root = cur->_right;
delete cur;
cur = NULL;
return true;
}
if (parent->_key > cur->_key)
{
del = cur;
parent->_left = cur->_right;
delete del;
return true;
}
else if (parent->_key < key)
{
del = cur;
parent->_right = cur->_right;
delete del;
return true;
}
}
//要删除节点的右子树为空，同样分3种情况
else if (cur->_right == NULL)
{
//注意判断父节点是否为空，若为空，则要删除的节点为根节点,如：只有根节点5和其左节点3
if (parent == NULL)
{
_root = cur->_left;
delete cur;
cur = NULL;
return true;
}
if (parent->_key > cur->_key)
{
del = cur;
parent->_left = cur->_left;
delete del;
return true;
}
else if (parent->_key < cur->_key)
{
del = cur;
parent->_right = cur->_left;
delete del;
return true;
}
}
//左右子树都不为空
else
{
Node* del = cur;
Node* parent = NULL;
Node* RightFirst = cur->_right;
//右边第一个节点的左子树为空
if (RightFirst->_left == NULL)
{
swap(RightFirst->_key, cur->_key);
swap(RightFirst->_value, cur->_value);
del = RightFirst;
cur->_right = RightFirst->_right;
delete del;
return true;
}
//右边第一个节点的左子树不为空
while (RightFirst->_left)
{
parent = RightFirst;
RightFirst = RightFirst->_left;
}
swap(RightFirst->_key, cur->_key);
swap(RightFirst->_value, cur->_value);
del = RightFirst;
parent->_left = RightFirst->_right;
delete del;
return true;
}
}
}
return false;
}

bool Insert_R(const K& key, const V& value)
{
return _Insert_R(_root, key, value);
}

Node* Find_R(const K& key)
{
return _Find_R(_root, key);
}

bool Remove_R(const K& key)
{
return _Remove_R(_root, key);
}

void InOrder()
{
_InOrder(_root);
cout << endl;
}

protected:

bool _Remove_R(Node*& root, const K& key)
{
//没有节点
if (root == NULL)
{
return false;
}
//只有一个节点
if (root->_left == NULL&&root->_right == NULL)
{
if (root->_key == key)
{
delete root;
root = NULL;
return true;
}
else
{
return false;
}

}

//删除二叉搜索树节点的递归写法
if (root->_key > key)
{
_Remove_R(root->_left, key);
}
else if (root->_key < key)
{
_Remove_R(root->_right, key);
}
else
{
Node* del = NULL;

if (root->_left == NULL)
{
del = root;
root = root->_right;
delete del;
del = NULL;
return true;
}
else if (root->_right == NULL)
{
del = root;
root = root->_left;
delete del;
del = NULL;
return true;
}
else
{
Node* RightFirst = root->_right;

while (RightFirst->_left)
{
RightFirst = RightFirst->_left;
}

swap(root->_key, RightFirst->_key);
swap(root->_value, RightFirst->_value);

_Remove_R(root->_right, key);
return true;
}
}
}

//递归查找法
Node* _Find_R(Node* root, const K& key)
{
if (root == NULL)
{
return NULL;
}
if (root->_key > key)
{
return _Find_R(root->_left, key);
}
else if (root->_key < key)
{
return _Find_R(root->_right, key);
}
else
{
return root;
}
}

//递归插入法
bool _Insert_R(Node*& root, const K& key, const V& value)
{
if (root == NULL)
{
root = new Node(key, value);
return true;
}
if (root->_key > key)
{
return _Insert_R(root->_left, key, value);
}
else if(root->_key < key)
{
return _Insert_R(root->_right, key, value);
}
else
{
return false;
}
}

void _InOrder(Node* root)
{
if (root == NULL)
{
return;
}

_InOrder(root->_left);
cout << root->_key << " ";
_InOrder(root->_right);
}
protected:
Node* _root;

};

void Test()
{
BSTree<int, int> s;

//测试插入
s.Insert_R(5, 1);
s.Insert_R(4, 1);
s.Insert_R(3, 1);
s.Insert_R(6, 1);
s.Insert_R(1, 1);
s.Insert_R(2, 1);
s.Insert_R(0, 1);
s.Insert_R(9, 1);
s.Insert_R(8, 1);
s.Insert_R(7, 1);

//二叉搜索树按中序输出是有序的
s.InOrder();

//测试查找
cout << s.Find_R(6)->_key << endl;

//测试删除
s.Remove(4);
s.Remove(6);
s.Remove(3);
s.Remove(1);
s.Remove(2);

//再次打印删除后的结果
s.InOrder();

}

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

运行结果：

0 1 2 3 4 5 6 7 8 9
6
0 5 7 8 9
请按任意键继续. . .