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后续有时间会增加erase

1.红黑树的概念

红黑树是一种自平衡的二叉搜索树。每个节点额外存储了一个 color 字段 (“RED” or “BLACK”)， 通过对任何一条从根到叶子的路径上各个结点着色方式的限制，红黑树确保没有一条路径会比其他路径长出俩倍，因而是接近平衡。

2.红黑树的性质

一棵合法的红黑树必须遵循以下四条性质：

1. 节点为红色或黑色
2. 根节点是黑色的 （在不同的实现下，该条性质并非必须满足）
3. NIL 节点（空叶子节点）为黑色
4. 红色节点的子节点为黑色
5. 从根节点到 NIL节点的每条路径上的黑色节点数量相同

3.红黑树的结点

enum Colour
{RED,BLACK
};template<class K, class V>
struct RBTreeNode
{RBTreeNode<K, V>* _left;RBTreeNode<K, V>* _right;RBTreeNode<K, V>* _parent;std::pair<K, V> _kv;Colour _col;RBTreeNode(const std::pair<K, V>& kv):_left(nullptr), _right(nullptr), _parent(nullptr), _kv(kv), _col(RED){}
};

4.insert函数（插入结点）

新节点的默认颜色是红色（如果其双亲节点的颜色是黑色，没有违反红黑树任何性质，则不需要调整；但当新插入节点的双亲节点颜色为红色时，就违反了性质4不能有连在一起的红色节点，此时需要对红黑树分情况来讨论）
约定:cur为当前节点，p为父节点，g为祖父节点，u为叔叔节点
所有插入的情况可分为以下三种：

• 情况一: cur为红，p为红，g为黑，u存在且为红
• 情况二: cur为红，p为红，g为黑，u不存在/u存在且为黑
• 情况三: cur为红，p为红，g为黑，u不存在/u存在且为黑

	bool Insert(const std::pair<K, V>& kv){if (_root == nullptr){_root = new Node(kv);_root->_col = BLACK;return true;}Node* parent = nullptr;Node* cur = _root;while (cur){if (cur->_kv.first < kv.first){parent = cur;cur = cur->_right;}else if (cur->_kv.first > kv.first){parent = cur;cur = cur->_left;}else{return false;}}cur = new Node(kv);cur->_col = RED; // 新增节点给红色if (parent->_kv.first < kv.first){parent->_right = cur;}else{parent->_left = cur;}cur->_parent = parent;// parent的颜色是黑色也结束while (parent && parent->_col == RED){// 关键看叔叔Node* grandfather = parent->_parent;if (parent == grandfather->_left){Node* uncle = grandfather->_right;// 叔叔存在且为红，-》变色即可if (uncle && uncle->_col == RED){parent->_col = uncle->_col = BLACK;grandfather->_col = RED;// 继续往上处理cur = grandfather;parent = cur->_parent;}else // 叔叔不存在，或者存在且为黑{if (cur == parent->_left){//     g  //   p   u// c RotateR(grandfather);parent->_col = BLACK;grandfather->_col = RED;}else{//      g  //   p     u//      c RotateL(parent);RotateR(grandfather);cur->_col = BLACK;grandfather->_col = RED;}break;}}else{Node* uncle = grandfather->_left;// 叔叔存在且为红，-》变色即可if (uncle && uncle->_col == RED){parent->_col = uncle->_col = BLACK;grandfather->_col = RED;// 继续往上处理cur = grandfather;parent = cur->_parent;}else // 叔叔不存在，或者存在且为黑{// 情况二：叔叔不存在或者存在且为黑// 旋转+变色//      g//   u     p//            cif (cur == parent->_right){RotateL(grandfather);parent->_col = BLACK;grandfather->_col = RED;}else{//		g//   u     p//      cRotateR(parent);RotateL(grandfather);cur->_col = BLACK;grandfather->_col = RED;}break;}}}_root->_col = BLACK;return true;}

5.左旋、右旋

这里和AVLTree的旋转一样

	void RotateR(Node* parent){Node* subL = parent->_left;Node* subLR = subL->_right;parent->_left = subLR;if (subLR)subLR->_parent = parent;subL->_right = parent;Node* ppNode = parent->_parent;parent->_parent = subL;if (parent == _root){_root = subL;_root->_parent = nullptr;}else{if (ppNode->_left == parent){ppNode->_left = subL;}else{ppNode->_right = subL;}subL->_parent = ppNode;}}void RotateL(Node* parent){Node* subR = parent->_right;Node* subRL = subR->_left;parent->_right = subRL;if (subRL)subRL->_parent = parent;subR->_left = parent;Node* ppNode = parent->_parent;parent->_parent = subR;if (parent == _root){_root = subR;_root->_parent = nullptr;}else{if (ppNode->_right == parent){ppNode->_right = subR;}else{ppNode->_left = subR;}subR->_parent = ppNode;}}

6.总代码

#include<vector>enum Colour
{RED,BLACK
};template<class K, class V>
struct RBTreeNode
{RBTreeNode<K, V>* _left;RBTreeNode<K, V>* _right;RBTreeNode<K, V>* _parent;std::pair<K, V> _kv;Colour _col;RBTreeNode(const std::pair<K, V>& kv):_left(nullptr), _right(nullptr), _parent(nullptr), _kv(kv), _col(RED){}
};template<class K, class V>
class RBTree
{typedef RBTreeNode<K, V> Node;
public:bool Insert(const std::pair<K, V>& kv){if (_root == nullptr){_root = new Node(kv);_root->_col = BLACK;return true;}Node* parent = nullptr;Node* cur = _root;while (cur){if (cur->_kv.first < kv.first){parent = cur;cur = cur->_right;}else if (cur->_kv.first > kv.first){parent = cur;cur = cur->_left;}else{return false;}}cur = new Node(kv);cur->_col = RED; // 新增节点给红色if (parent->_kv.first < kv.first){parent->_right = cur;}else{parent->_left = cur;}cur->_parent = parent;// parent的颜色是黑色也结束while (parent && parent->_col == RED){// 关键看叔叔Node* grandfather = parent->_parent;if (parent == grandfather->_left){Node* uncle = grandfather->_right;// 叔叔存在且为红，-》变色即可if (uncle && uncle->_col == RED){parent->_col = uncle->_col = BLACK;grandfather->_col = RED;// 继续往上处理cur = grandfather;parent = cur->_parent;}else // 叔叔不存在，或者存在且为黑{if (cur == parent->_left){//     g  //   p   u// c RotateR(grandfather);parent->_col = BLACK;grandfather->_col = RED;}else{//      g  //   p     u//      c RotateL(parent);RotateR(grandfather);cur->_col = BLACK;grandfather->_col = RED;}break;}}else{Node* uncle = grandfather->_left;// 叔叔存在且为红，-》变色即可if (uncle && uncle->_col == RED){parent->_col = uncle->_col = BLACK;grandfather->_col = RED;// 继续往上处理cur = grandfather;parent = cur->_parent;}else // 叔叔不存在，或者存在且为黑{// 情况二：叔叔不存在或者存在且为黑// 旋转+变色//      g//   u     p//            cif (cur == parent->_right){RotateL(grandfather);parent->_col = BLACK;grandfather->_col = RED;}else{//		g//   u     p//      cRotateR(parent);RotateL(grandfather);cur->_col = BLACK;grandfather->_col = RED;}break;}}}_root->_col = BLACK;return true;}void RotateR(Node* parent){Node* subL = parent->_left;Node* subLR = subL->_right;parent->_left = subLR;if (subLR)subLR->_parent = parent;subL->_right = parent;Node* ppNode = parent->_parent;parent->_parent = subL;if (parent == _root){_root = subL;_root->_parent = nullptr;}else{if (ppNode->_left == parent){ppNode->_left = subL;}else{ppNode->_right = subL;}subL->_parent = ppNode;}}void RotateL(Node* parent){Node* subR = parent->_right;Node* subRL = subR->_left;parent->_right = subRL;if (subRL)subRL->_parent = parent;subR->_left = parent;Node* ppNode = parent->_parent;parent->_parent = subR;if (parent == _root){_root = subR;_root->_parent = nullptr;}else{if (ppNode->_right == parent){ppNode->_right = subR;}else{ppNode->_left = subR;}subR->_parent = ppNode;}}void InOrder(){_InOrder(_root);cout << endl;}bool IsBalance(){if (_root->_col == RED){return false;}int refNum = 0;Node* cur = _root;while (cur){if (cur->_col == BLACK){++refNum;}cur = cur->_left;}return Check(_root, 0, refNum);}private:bool Check(Node* root, int blackNum, const int refNum){if (root == nullptr){//cout << blackNum << endl;if (refNum != blackNum){cout << "存在黑色节点的数量不相等的路径" << endl;return false;}return true;}if (root->_col == RED && root->_parent->_col == RED){cout << root->_kv.first << "存在连续的红色节点" << endl;return false;}if (root->_col == BLACK){blackNum++;}return Check(root->_left, blackNum, refNum)&& Check(root->_right, blackNum, refNum);}void _InOrder(Node* root){if (root == nullptr){return;}_InOrder(root->_left);cout << root->_kv.first << ":" << root->_kv.second << endl;_InOrder(root->_right);}private:Node* _root = nullptr;//size_t _size = 0;
};