1.概念

1.父节点的左子树全小于本身

2.父节点的右子树全大于本身

3.左右子树也是二叉搜索树

时间复杂度：O(N)，有可能只有左数，这样就遍历了所有，所有复杂度为N

平衡二叉树的时间复杂度才是：O(logN)

2.模拟

1.数据元素设计

1.一个空间存储数据date

2.两个指针，用于指向左右子树的位置

template<class K>
struct BSTreeNode
{BSTreeNode(const K& key):_key(key),_left(nullptr),_right(nullptr){}BSTreeNode<K>* _left;BSTreeNode<K>* _right;K _key;
};

2. 搜索二叉树类的定义

template<class K>
class BSTree
{
private:Node* _root = nullptr;
};

3.构造函数，拷贝构造，赋值拷贝

1.构造函数，直接定义空树

2.拷贝构造，取调用私有的copy函数

3.赋值拷贝，直接swap两个地址

copy函数的实现，其实是递归拷贝

1.确认传入参数是二叉搜索树的指针；返回返回头指针

2.如果指向的指针是空，则返回

3.构造函数后再考虑左右，所以实现是靠前序遍历

4.一个节点为单位，指向有数据的节点，则new一个新的指针，数据拷贝

5.new出来的子节点，左树由copy函数构建，右树也由copy函数构建

BSTree():_root(nullptr)
{}BSTree(const BSTree<K>& t)
{_root = Copy(t._root);
}BSTree<K>& operator=(BSTree<K> t)
{swap(_root, t._root);return *this;
}private:
Node* Copy(Node* root)
{if (root == nullptr)return nullptr;Node* newroot = new Node(root->_key);newroot->_left = Copy(root->_left);newroot->_right = Copy(root->_right);return newroot;
}

4.析构函数

析构函数，调用Destory函数，再把_root指向空

相同的，Destory也可以通过递归销毁

1.确认传入参数是二叉搜索树的指针；由于销毁所以返回的值不需要

2.如果指向的指针是空，则返回

3.前面的销毁，后面就找不到了；所以我们要到最后再销毁最开始的指针，所以用后后续遍历

4.Destory左右子树

5.走到最后，要销毁指针

~BSTree()
{Destory(_root);_root = nullptr;
}private:
void Destory(Node* root)
{if (root == nullptr)return;Destory(root->_left);Destory(root->_right);delete root;
}

5.二分法实现增删查

由于左小右大，完美符合二分法的实现

Insert

1.如果为空，则更新_root，把节点的数据存储k；如果不为空向下走

2.临时变量cur和parent指针，cur指向开头；随后比较，小左大右；更新parent为旧的cur

3.如果夹在中间的k跟parent与cur存在逻辑矛盾返回false

4.k夹在parent和cur之间，cur位置就是k要存储的位置；cur重新new成数据为k的节点，通过parent指针找到原先的cur，中间插入即可，随后返回true

Find：直接找，二分法，找到了返回true，找不到返回false

Erase：先找到该k数据对于节点，随后请看代码的注释

bool Insert(const K& k)
{if (_root == nullptr){_root = new Node(k);return true;}Node* cur = _root;Node* parent = nullptr;while (cur){if (cur->_key < k){parent = cur;cur = cur->_right;}else if (cur->_key > k){parent = cur;cur = cur->_left;}elsereturn false;}cur = new Node(k);if (parent->_key > k)parent->_left = cur;elseparent->_right = cur;return true;
}bool Find(const K& k)
{Node* cur = _root;while (cur){if (cur->_key < k)cur = cur->_right;else if (cur->_key > k)cur = cur->_left;elsereturn true;}return false;
}bool Erase(const K& k)
{Node* cur = _root;Node* parent = nullptr;while (cur){if (cur->_key < k){parent = cur;cur = cur->_right;}else if (cur->_key > k){parent = cur;cur = cur->_left;}else{//删除节点的左子树为空//父节点连接节点的右字数if (cur->_left == nullptr){if (cur == _root)  // 删除的1，2两种情况下，解决parent为空的情况_root = cur->_right;else{if (parent->_key > k) //说明cur在parent的左边parent->_left = cur->_right;elseparent->_right = cur->_right;}delete cur;return true;}//删除节点的右子树为空//父节点连接节点的左字数else if (cur->_right == nullptr){if (cur == _root)_root = cur->_left;else{if (parent->_key > k) //说明cur在parent的左边parent->_left = cur->_left;elseparent->_right = cur->_left;}delete cur;return true;}else{Node* pur = cur;Node* tmp = Get_Max(cur->_left, pur);cur->_key = tmp->_key;if (pur->_right == tmp)pur->_right = tmp->_left;elsepur->_left = tmp->_right;delete tmp;return true;}}}return false;
}

6.递归实现增删查

bool _InsertR(Node*& root, const K& k){if (root==nullptr){root = new Node(k);return true;}if (root->_key < k)return _InsertR(root->_right, k);else if (root->_key > k)return _InsertR(root->_left, k);elsereturn false;}bool _Find(Node* root, const K& k){if (root == nullptr)return false;if (root->_key > k)_Find(root->_left, k);else if (root->_key < k)_Find(root->_left, k);elsereturn true;}bool _Erase(Node*& root, const K& k){if (root == nullptr)return false;if (root->_key > k)_Erase(root->_left, k);else if (root->_key < k)_Erase(root->_right, k);else{Node* del = root;if (root->_left == nullptr)root = root->_right;else if(root->_right == nullptr)root = root->_left;else{Node* minRight = root->_right;while (minRight->_left){minRight = minRight->_left;}swap(root->_key, minRight->_key);return _Erase(root->_right, k);}delete del;return true;}}

3.整体实现

template<class K>
struct BSTreeNode
{BSTreeNode(const K& key):_key(key),_left(nullptr),_right(nullptr){}BSTreeNode<K>* _left;BSTreeNode<K>* _right;K _key;
};template<class K>
class BSTree
{typedef BSTreeNode<K> Node;
public:BSTree():_root(nullptr){}BSTree(const BSTree<K>& t){_root = Copy(t._root);}BSTree<K>& operator=(BSTree<K> t){swap(_root, t._root);return *this;}~BSTree(){Destory(_root);_root = nullptr;}bool Insert(const K& k){if (_root == nullptr){_root = new Node(k);return true;}Node* cur = _root;Node* parent = nullptr;while (cur){if (cur->_key < k){parent = cur;cur = cur->_right;}else if (cur->_key > k){parent = cur;cur = cur->_left;}elsereturn false;}cur = new Node(k);if (parent->_key > k)parent->_left = cur;elseparent->_right = cur;return true;}bool Find(const K& k){Node* cur = _root;while (cur){if (cur->_key < k)cur = cur->_right;else if (cur->_key > k)cur = cur->_left;elsereturn true;}return false;}bool Erase(const K& k){Node* cur = _root;Node* parent = nullptr;while (cur){if (cur->_key < k){parent = cur;cur = cur->_right;}else if (cur->_key > k){parent = cur;cur = cur->_left;}else{//删除节点的左子树为空//父节点连接节点的右字数if (cur->_left == nullptr){if (cur == _root)  // 删除的1，2两种情况下，解决parent为空的情况_root = cur->_right;else{if (parent->_key > k) //说明cur在parent的左边parent->_left = cur->_right;elseparent->_right = cur->_right;}delete cur;return true;}//删除节点的右子树为空//父节点连接节点的左字数else if (cur->_right == nullptr){if (cur == _root)_root = cur->_left;else{if (parent->_key > k) //说明cur在parent的左边parent->_left = cur->_left;elseparent->_right = cur->_left;}delete cur;return true;}else{Node* pur = cur;Node* tmp = Get_Max(cur->_left, pur);cur->_key = tmp->_key;if (pur->_right == tmp)pur->_right = tmp->_left;elsepur->_left = tmp->_right;delete tmp;return true;}}}return false;}void Print(){_Print(_root);cout << endl;}/*bool InsertR(const K& k){if (_root == nullptr){_root = new Node(k);return true;}Node* parent = nullptr;return _InsertR(_root, k, parent);}*/bool InsertR(const K& k){return _InsertR(_root, k);}bool FindR(const K& k){return _Find(_root, k);}bool EraseR(const K& k){return _Erase(_root, k);}private:Node* Copy(Node* root){if (root == nullptr)return nullptr;Node* newroot = new Node(root->_key);newroot->_left = Copy(root->_left);newroot->_right = Copy(root->_right);return newroot;}/*bool _InsertR(Node* root, const K& k, Node* &parent){Node* cur = root;if (cur != nullptr){if (cur->_key > k){parent = cur;_InsertR(cur->_left, k, parent);}else if (cur->_key < k){parent = cur;_InsertR(cur->_right, k, parent);}}else{cur = new Node(k);if (parent->_key > k)parent->_left = cur;elseparent->_right = cur;return true;}if (root->_key == k)return false;}*///进阶思路----引用的使用bool _InsertR(Node*& root, const K& k){if (root==nullptr){root = new Node(k);return true;}if (root->_key < k)return _InsertR(root->_right, k);else if (root->_key > k)return _InsertR(root->_left, k);elsereturn false;}bool _Find(Node* root, const K& k){if (root == nullptr)return false;if (root->_key > k)_Find(root->_left, k);else if (root->_key < k)_Find(root->_left, k);elsereturn true;}bool _Erase(Node*& root, const K& k){if (root == nullptr)return false;if (root->_key > k)_Erase(root->_left, k);else if (root->_key < k)_Erase(root->_right, k);else{Node* del = root;if (root->_left == nullptr)root = root->_right;else if(root->_right == nullptr)root = root->_left;else{Node* minRight = root->_right;while (minRight->_left){minRight = minRight->_left;}swap(root->_key, minRight->_key);return _Erase(root->_right, k);}delete del;return true;}}void Destory(Node* root){if (root == nullptr)return;Destory(root->_left);Destory(root->_right);delete root;}Node* Get_Max(Node* root, Node* &parent){while (root->_right){parent = root;root = root->_right;}return root;}void _Print(Node* root){if (root == nullptr)return;_Print(root->_left);cout << root->_key << " ";_Print(root->_right);}private:Node* _root = nullptr;
};