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;   }  }

        //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->_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;}

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