红黑树的概念

红黑树，是一种二叉搜索树，但在每个结点上增加一个存储位表示结点的颜色，可以是Red或 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两倍，因而是接近平衡的。

红黑树性质

1. 每个结点不是红色就是黑色

2. 根节点是黑色的

3. 如果一个节点是红色的，则它的两个孩子结点是黑色的（也就是没有连续的红色节点）

4. 对于每个结点，从该结点到其所有后代叶结点的简单路径上，均包含相同数目的黑色结点

5. 每个叶子结点都是黑色的(此处的叶子结点指的是空结点)

红黑树节点设计

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

红黑树的插入

红黑树是在二叉搜索树的基础上加上其平衡限制条件，因此红黑树的插入可分为两步：

1. 按照二叉搜索的树规则插入新节点

2. 检测新节点插入后，红黑树的性质是否造到破坏

因为新节点的默认颜色是红色，因此：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，就违反了性质三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：

完整插入代码如下

bool Insert(const pair<K,V>& kv) {
		if (_pHead == nullptr) {
			_pHead = new Node(kv);
			_pHead->col = BLACK;
			return true;
		}
		Node* parent = nullptr;
		Node* cur = _pHead;
		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;
		if (parent->_kv.first < kv.first) {
			parent->_right = cur;
			cur->_parent = parent;
		}
		else {
			parent->_left = cur;
			cur->_parent parent;
		}

		//控制平衡
		while (parent && parent->_col == RED) {
			Node* grandFather = parent->_parent;
			if (parent == grandFather->_left) {
				Node* uncle = grandFather->_right;
				if (uncle && uncle->_col == RED) {//uncle存在且为红
					parent->_col = uncle->_col = BLACK;
					grandFather->_col = RED;
					cur = grandFather;
					parent = cur->_parent;
				}
				else {//uncle不存在或者存在且黑
					if (cur == parent->_left) {
						//右单旋
						RotateR(grandFather);
						parent->_col = BLACK;
						grandFather->_col = RED;
					}
					else {
						RotateL(parent);
						RotateR(grandFather);
						grandFather->_col = RED;
						cur->_col = BLACK;
					}
					break;
				}
			}
			else {//parent == grandFather->_right
				Node* uncle = grandFather->_left;
				if (uncle && uncle->_col == RED) {
					parent->_col = uncle->_col = BLACK;
					grandFather->_col = RED;
					cur = grandFather;
					parent = cur->_parent;
				}
				else {//uncle不存在或者存在且为黑
					if (cur == parent->_right ) {
						//左单旋
						RotateL(grandFather);
						parent->_col = BLACK;
						grandFather->_col = RED;
					}
					else {
						RotateR(parent);
						RotateL(grandFather);
						grandFather->_col = RED;
						cur->_col = BLACK;
					}
					break;
				}
			}
		}
		_pHead->_col = BLACK;
		return true;
	}

红黑树的验证

红黑树的检测分为两步：

1. 检测其是否满足二叉搜索树(中序遍历是否为有序序列)

2. 检测其是否满足红黑树的性质

    bool IsValidRBTree() {
		if (_pHead->_col == RED) {
			cout << "根节点不是黑色" << endl;
			return false;
		}
		int banchMark = 0;
		Node* left = _pHead;
		while (left) {
			if (left->_col == BLACK) {
				banchMark++;
			}
			left = left->_left;
		}
		int blackNum = 0;
		return _IsValidRBTree(_pHead, blackNum, banchMark);
	}
	void _InOrder(Node* root) {
		if (root == nullptr) {
			return;
		}
		_InOrder(root->_left);
		cout << root->_kv.first << ":" << root->_kv.second << endl;
		_InOrder(root->_right);
	}
	void InOrder() {
		_InOrder(_pHead);
	}
private:
	bool _IsValidRBTree(Node* root, size_t blackCount, size_t pathBlack) {
		if (root == nullptr) {
			if (blackCount != pathBlack) {
				cout << "存在路径黑色节点的数量不相等" << endl;
				return false;
			}
			return true;
		}
		if (root->_col == RED && root->_parent->_col == RED) {
			cout << "出现连续红色节点" << endl;
			return false;
		}
		if (root->_col == BLACK) {
			blackCount++;
		}
		return _IsValidRBTree(root->_left, blackCount, pathBlack)
			&& _IsValidRBTree(root->_right, blackCount, pathBlack);
	}