1. 二叉搜索树的实现

namespace key
{template<class K>struct BSTreeNode{typedef BSTreeNode<K> Node;Node* _left;Node* _right;K _key;BSTreeNode(const K& key): _left(nullptr), _right(nullptr), _key(key){}};template<class K>class BSTree{typedef BSTreeNode<K> Node;public:// 强制生成默认构造BSTree() = default;BSTree(const BSTree<K>& t){_root = Copy(t._root);}BSTree<K>& operator=(BSTree<K> t){swap(_root, t._root);return *this;}~BSTree(){Destroy(_root);}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;}bool 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 true;}}return false;}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{if (cur->_left == nullptr){if (cur == _root){_root = cur->_right;}else{if (cur == parent->_right){parent->_right = cur->_right;}else{parent->_left = cur->_right;}}delete cur;return true;}else if (cur->_right == nullptr){if (cur == _root){_root = cur->_left;}else{if (cur == parent->_right){parent->_right = cur->_left;}else{parent->_left = cur->_left;}}delete cur;return true;}else{// 替换法Node* rightMinParent = cur;Node* rightMin = cur->_right;while (rightMin->_left){rightMinParent = rightMin;rightMin = rightMin->_left;}cur->_key = rightMin->_key;if (rightMin == rightMinParent->_left){rightMinParent->_left = rightMin->_right;}else{rightMinParent->_right = rightMin->_right;}delete rightMin;return true;}}}return false;}void InOrder(){_InOrder(_root);cout << endl;}bool FindR(const K& key){return _FindR(_root, key);}bool InsertR(const K& key){return _InsertR(_root, key);}bool EraseR(const K& key){return _EraseR(_root, key);}private:void Destroy(Node* root){if (root == nullptr)return;Destroy(root->_left);Destroy(root->_right);delete root;}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;}void _InOrder(Node* root){if (root == nullptr)return;_InOrder(root->_left);cout << root->_key << " ";_InOrder(root->_right);}bool _FindR(Node* root, const K& key){if (root == nullptr)return false;if (root->_key < key){return _FindR(root->_right, key);}else if (root->_key > key){return _FindR(root->_left, key);}else{return true;}}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{Node* del = root;if (root->_right == nullptr){root = root->_left;}else if (root->_left == nullptr){root = root->_right;}else{Node* rightMin = root->_right;while (rightMin->_left){rightMin = rightMin->_left;}swap(root->_key, rightMin->_key);return _EraseR(root->_right, key);}delete del;return true;}}private:Node* _root = nullptr;};
}

2. 二叉搜索树的应用

1. 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：

   • 以词库中所有单词集合中的每个单词作为 key，构建一颗二叉搜索树；
   • 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。

2. KV 模型：每一个关键码 key，都有与之对应的值 value，即 <Key, Value> 的键值对。这种方式在现实生活中非常常见：

   • 比如英汉词典就是英文与中文的对应关系，通过英文可以快速找到与其对应的中文，英文单词与其对应的中文 <word, chinese> 就构成一种键值对；
   • 再比如统计单词个数，统计成功后，给定单词就可以快速找到其出现的次数，单词与其出现次数 <word, count> 就构成一种键值对。

3. 改造二叉搜索树为 KV 结构

namespace key_value
{template<class K, class V>struct BSTreeNode{typedef BSTreeNode<K, V> Node;Node* _left;Node* _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;public:bool Insert(const K& key, const V& value){if (_root == nullptr){_root = new Node(key, value);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, value);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 nullptr;}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{if (cur->_left == nullptr){if (cur == _root){_root = cur->_right;}else{if (cur == parent->_right){parent->_right = cur->_right;}else{parent->_left = cur->_right;}}delete cur;return true;}else if (cur->_right == nullptr){if (cur == _root){_root = cur->_left;}else{if (cur == parent->_right){parent->_right = cur->_left;}else{parent->_left = cur->_left;}}delete cur;return true;}else{// 替换法Node* rightMinParent = cur;Node* rightMin = cur->_right;while (rightMin->_left){rightMinParent = rightMin;rightMin = rightMin->_left;}cur->_key = rightMin->_key;if (rightMin == rightMinParent->_left){rightMinParent->_left = rightMin->_right;}else{rightMinParent->_right = rightMin->_right;}delete rightMin;return true;}}}return false;}void InOrder(){_InOrder(_root);cout << endl;}private:void _InOrder(Node* root){if (root == nullptr)return;_InOrder(root->_left);cout << root->_key << " ";cout << root->_value << endl;_InOrder(root->_right);}private:Node* _root = nullptr;};
}

测试代码：

namespace key_value
{void TestBSTree1(){// 输入单词，查找单词对应的中文翻译BSTree<string, string> dict;dict.Insert("string", "字符串");dict.Insert("tree", "树");dict.Insert("left", "左边、剩余");dict.Insert("right", "右边");dict.Insert("sort", "排序");// 插入词库中所有单词string str;while (cin >> str){BSTreeNode<string, string>* ret = dict.Find(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 << "中文翻译:" << ret->_value << endl;}}}void TestBSTree2(){// 统计水果出现的次数string arr[] = { "苹果", "西瓜", "苹果", "西瓜", "苹果", "苹果", "西瓜","苹果", "香蕉", "苹果", "香蕉" };BSTree<string, int> countTree;for (const auto& str : arr){// 先查找水果在不在搜索树中// 1、不在，说明水果第一次出现，则插入<水果, 1>// 2、在，则查找到的节点中水果对应的次数++//BSTreeNode<string, int>* ret = countTree.Find(str);auto ret = countTree.Find(str);if (ret == NULL){countTree.Insert(str, 1);}else{ret->_value++;}}countTree.InOrder();}
}

4. 二叉搜索树的性能分析

插入和删除操作都必须先查找，查找效率代表了二叉搜索树中各个操作的性能。

对有 n 和节点的二叉搜索树，若每个元素查找的概率相等，则二叉搜索树平均查找长度是节点在二叉搜索树的深度的函数，即节点越深，比较次数越多。

但对于同一个关键码集合，如果各关键码插入的次序不同，可能得到不同结构的二叉搜索树：

最优情况下，二叉搜索树为完全二叉树（或者接近完全二叉树），其平均比较次数为：$lo{g}_{2}N$；

最差情况下，二叉搜索树退化为单支树（或者类似单支），其平均比较次数为：$\frac{N}{2}$。

问题：如果退化成单支树，二叉搜索树的性能就失去了。能否进行改进，不论按照什么次序插入关键码，都能使二叉搜索树的性能达到最优？

答：可以使用 AVL树 和 红黑树 的特性，使单支树在达到一定深度时进行“旋转”，变回接近完全二叉树的形状，这个我们后面再谈。

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