C++二叉树进阶
二叉树进阶
- 二叉搜索树
-
- 二叉搜索树概念
- 二叉树的操作
-
- 插入
- 查找
- 删除
- 遍历(中序)
- 整体实现
- 搜索二叉树的应用
-
- 整体实现
二叉搜索树
二叉搜索树概念
二叉搜索树又称二叉排序树，它或者是一棵空树，或者是具有以下性质的二叉树:
- 若它的左子树不为空，则左子树上所有节点的值都小于根节点的值
- 若它的右子树不为空，则右子树上所有节点的值都大于根节点的值
- 它的左右子树也分别为二叉搜索树
时间复杂度：O(N)，只有当树的形状接近完全二叉树或者满二叉树，才能达到 logN
搜索二叉树延伸：AVLTree 红黑树(对搜索二叉树左右高度提出要求，非常接近完全二叉树，效率可达到 O(logN))
上面一般用于在内存中查找，当数据在磁盘中时，对树高度进一步提出要求-》衍生B树系列
二叉树的操作
插入
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;
}
查找
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 NULL;
}
删除
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
{
//找到，准备删除
//左为空或右为空，可直接删除，把另一个孩子交给父亲管理，删除自己
if (cur->_left == nullptr)//左为空
{
if (cur == _root)
{
_root = cur->_right;
}
else
{
if (parent->_left == cur)
{
parent->_left = cur->_right;
}
else
{
parent->_right = cur->_right;
}
}
delete cur;
}
else if (cur->_right == nullptr)//右为空
{
if (cur == _root)
{
_root = cur->_left;
}
else
{
if (parent->_left == cur)
{
parent->_left = cur->_left;
}
else
{
parent->_right = cur->_left;
}
}
}
else//左右都不为空，替换法删除
{
找到右子树最小的节点去替换
//Node* minparent = cur;
//Node* minRight = cur->_right;
//while (minRight->_left)
//{
// minparent = minRight;
// minRight = minRight->_left;
//}
保存替换节点
//cur->_key = minRight->_key;
删除替换节点
//if (minparent->_left == minRight)
//{
// minparent->_left = minRight->_right;
//}
//else
//{
// minparent->_right = minRight->_right;
//}
//delete minRight;
Node* minRight = cur->_right;
while (minRight->_left)
{
minRight = minRight->_left;
}
K min = minRight->_key;
//递归调用自己去删除替换节点
this->Erase(min);
cur->_key = min;
}
return true;
}
}
return false;
}
遍历(中序)
void _Inorder(Node* root)
{
if (root == nullptr)
{
return;
}
_Inorder(root->_left);
cout << root->_key << endl;
_Inorder(root->_right);
}
void Inorder()
{
_Inorder(_root);
cout << endl;
}
整体实现
#include
using namespace std;
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;
private:
Node* _FindR(Node* root,const K& key)
{
if (root == nullptr)
{
return nullptr;
}
if (root->_key < key)
{
return _FindR(root->_right, key);
}
else if (root->_key > key)
{
return _FindR(root->_left, key);
}
else
{
return root;
}
}
bool _InsertR(Node*& root, const K& key)
{
if (root == nullptr)
{
root = new Node(key);
return true;
}
if (root->_key < key)
{
return _InsertR(root->_right, key);
}
else if (root->_key > key)
{
return _InsertR(root->_left, key);
}
else
{
return false;
}
}
bool _EraseR(Node*& root, const K& key)
{
if (root == nullptr)
{
return false;
}
if (root->_key < key)
{
return _EraseR(root->_right, key);
}
else if (root->_key > key)
{
return _EraseR(root->_left, key);
}
else
{
//找到了,root就是要删除的节点
if (root->_left == nullptr)
{
Node* del = root;
root = root->_right;
delete del;
}
else if (root->_right == nullptr)
{
Node* del = root;
root = root->_left;
delete del;
}
else
{
找右子树最小节点
//Node* minParent = root;
//Node* minRight = root->_right;
//while (minRight->_left)
//{
// minParent = minRight;
// minRight = minRight->_left;
//}
保存替换节点的值
//root->key = minRight->_key;
//if (minParent->_left == minRight)
//{
// minParent->_left = minRight->_right;
//}
//else
//{
// minParent->_right = minRight->_right;
//}
//delete minRight;
Node* minRight = root->_right;
while (minRight->_left)
{
minRight = minRight->_left;
K min = minRight->_key;
}
_EraseR(root->_right, min);
root->_key = min;
}
return ture;
}
}
void _Destory(Node* root)
{
if (root == NULL)
{
return;
}
_Destory(root->_left);
_Destory(root->_right);
delete root;
}
Node* _Copy(Node* root)
{
if (root == nullptr)
{
return nullptr;
}
Node* copyNode = new Node(root->_key);
copyNode->_left = _Copy(root->_left);
copyNode->_right = _Copy(root->_right);
return copyNode;
}
public:
BSTree()
:_root(nullptr)
{}
BSTree(const BSTree<K>& t)
{
_root = _Copy(t._root);
}
~BSTree()
{
_Destory(_root);
_root = nullptr;
}
BSTree<K>& operator=(BSTree<K> t)
{
swap(_root, t._root);
return *this;
}
bool InsertR(const K& key)//递归版本
{
return _InsertR(_root, key);
}
Node* FindR(const K& key)//递归版本
{
return _FindR(_root, key);
}
bool EraseR(const K& key)//递归版本
{
return _EraseR(_root, key);
}
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;
}
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 NULL;
}
void _Inorder(Node* root)
{
if (root == nullptr)
{
return;
}
_Inorder(root->_left);
cout << root->_key << endl;
_Inorder(root->_right);
}
void Inorder()
{
_Inorder(_root);
cout << endl;
}
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
{
//找到，准备删除
//左为空或右为空，可直接删除，把另一个孩子交给父亲管理，删除自己
if (cur->_left == nullptr)//左为空
{
if (cur == _root)
{
_root = cur->_right;
}
else
{
if (parent->_left == cur)
{
parent->_left = cur->_right;
}
else
{
parent->_right = cur->_right;
}
}
delete cur;
}
else if (cur->_right == nullptr)//右为空
{
if (cur == _root)
{
_root = cur->_left;
}
else
{
if (parent->_left == cur)
{
parent->_left = cur->_left;
}
else
{
parent->_right = cur->_left;
}
}
}
else//左右都不为空，替换法删除
{
找到右子树最小的节点去替换
//Node* minparent = cur;
//Node* minRight = cur->_right;
//while (minRight->_left)
//{
// minparent = minRight;
// minRight = minRight->_left;
//}
保存替换节点
//cur->_key = minRight->_key;
删除替换节点
//if (minparent->_left == minRight)
//{
// minparent->_left = minRight->_right;
//}
//else
//{
// minparent->_right = minRight->_right;
//}
//delete minRight;
Node* minRight = cur->_right;
while (minRight->_left)
{
minRight = minRight->_left;
}
K min = minRight->_key;
//递归调用自己去删除替换节点
this->Erase(min);
cur->_key = min;
}
return true;
}
}
return false;
}
private:
Node* _root;
};
搜索二叉树的应用
-
K模型：K模型即只有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，具体方式如下： 以单词集合中的每个单词作为key，构建一棵二叉搜索树在二叉搜索树中检索该单词是否存在，存在则拼写正确，不存在则拼写错误。
-
.KV模型：每一个关键码key，都有与之对应的值Value，即
查找
void test1()
{
KV::BSTree<string, string>dict;
dict.InsertR("string", "字符串");
dict.InsertR("tree", "树");
dict.InsertR("left", "左边，剩余");
dict.InsertR("right", "右边");
dict.InsertR("sort", "排序");
//插入词库中单词
string str;
while(cin >> str)
{
KV::BSTreeNode<string, string>* ret = dict.FindR(str);
if (ret == nullptr)
{
cout << "单词拼写错误，词库中没有这个单词" << str << endl;
}
else
{
cout << str << "->" << ret->_value << endl;
}
}
}
int main()
{
test1();
return 0;
}
统计出现的次数
void test2()
{
//统计水果出现的次数
string arr[] = { "苹果", "桃子", "香蕉", "桃子", "苹果", "西瓜", "桃子" };
KV::BSTree<string, int> countfriut;
for (const auto e : arr)
{
//先查找在不在搜索树里
//1、不在-》插入<水果，1>
//2、在-》
//KV::BSTreeNode* ret = countfriut.FindR(e);
auto ret = countfriut.FindR(e);
if (ret == nullptr)
{
countfriut.InsertR(e,1);
}
else
{
ret->_value++;
}
}
countfriut.Inorder();
}
int main()
{
test2();
return 0;
}
整体实现
namespace KV
{
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;
private:
Node* _FindR(Node* root, const K& key)
{
if (root == nullptr)
{
return nullptr;
}
if (root->_key < key)
{
return _FindR(root->_right, key);
}
else if (root->_key > key)
{
return _FindR(root->_left, key);
}
else
{
return root;
}
}
bool _InsertR(Node*& root, const K& key,const V& value)
{
if (root == nullptr)
{
root = new Node(key,value);
return true;
}
if (root->_key < key)
{
return _InsertR(root->_right, key,value);
}
else if (root->_key > key)
{
return _InsertR(root->_left, key,value);
}
else
{
return false;
}
}
bool _EraseR(Node*& root, const K& key)
{
if (root == nullptr)
{
return false;
}
if (root->_key < key)
{
return _EraseR(root->_right, key);
}
else if (root->_key > key)
{
return _EraseR(root->_left, key);
}
else
{
//找到了,root就是要删除的节点
if (root->_left == nullptr)
{
Node* del = root;
root = root->_right;
delete del;
}
else if (root->_right == nullptr)
{
Node* del = root;
root = root->_left;
delete del;
}
else
{
Node* minRight = root->_right;
while (minRight->_left)
{
minRight = minRight->_left;
K kmin = minRight->_key;
V vmin = minRight->_value;
}
_EraseR(root->_right, kmin);
root->_key = kmin;
root->_value = vmin;
}
return ture;
}
}
void _Destory(Node* root)
{
if (root == NULL)
{
return;
}
_Destory(root->_left);
_Destory(root->_right);
delete root;
}
Node* _Copy(Node* root)
{
if (root == nullptr)
{
return nullptr;
}
Node* copyNode = new Node(root->_key,root->_value);
copyNode->_left = _Copy(root->_left);
copyNode->_right = _Copy(root->_right);
return copyNode;
}
public:
BSTree()
:_root(nullptr)
{}
BSTree(const BSTree<K,V>& t)
{
_root = _Copy(t._root);
}
~BSTree()
{
_Destory(_root);
_root = nullptr;
}
BSTree<K,V>& operator=(BSTree<K,V> t)
{
swap(_root, t._root);
return *this;
}
bool InsertR(const K& key,const V& value)//递归版本
{
return _InsertR(_root, key,value);
}
Node* FindR(const K& key)//递归版本
{
return _FindR(_root, key);
}
bool EraseR(const K& key)//递归版本
{
return _EraseR(_root, key);
}
void _Inorder(Node* root)
{
if (root == nullptr)
{
return;
}
_Inorder(root->_left);
cout << root->_key<<":"<<root->_value << endl;
_Inorder(root->_right);
}
void Inorder()
{
_Inorder(_root);
cout << endl;
}
private:
Node* _root;
};
}