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。

红黑树的性质

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
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)

红黑树节点的定义

//节点的颜色
enum Color
{RED,BLACK
};template<class K, class V>
struct RBTreeNode
{RBTreeNode<K, V>* _left;  //左孩子RBTreeNode<K, V>* _right; //右孩子RBTreeNode<K, V>* _parent;//双亲pair<K, V> _kv;Color _col; //红黑颜色限制, 新插入节点默认为红色RBTreeNode(const pair<K, V>& kv, Color col = RED): _left(nullptr), _right(nullptr), _parent(nullptr), _kv(kv), _col(col) {}
};

红黑树的结构

为了后续实现关联式容器简单，红黑树的实现中增加一个头结点，因为跟节点必须为黑色，为了与根节点进行区分，将头结点给成黑色，并且让头结点的 _parent 域指向红黑树的根节点，_left域指向红黑树中最小的节点，_right域指向红黑树中最大的节点，如下：

红黑树的插入

红黑树是在二叉搜索树的基础上加上其颜色限制条件，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：

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

• 情况一: cur为红，p为红，g为黑，u存在且为红

如果 g是根节点，调整完成后，需要将 g改为红色
如果 g是子树，g一定有双亲，且 g的双亲如果是红色，需要继续向上调整

解决方式：将p，u改为黑，g改为红，然后把 g当成 cur，继续向上调整

• 情况二: cur为红，p为红，g为黑，u不存在/u为黑

u的情况有两种：
(1)如果 u节点不存在，则 cur一定是新插入节点，因为如果 cur不是新插入节点，则 cur和 p一定有一个节点的颜色是黑色，就不满足性质4：每条路径黑色节点个数相同。
(2)如果 u节点存在，则其一定是黑色的，那么 cur节点原来的颜色一定也是黑色的，现在看到其是红色的原因是因为 cur的子树在调整的过程中将 cur节点的颜色由黑色改成红色。

解决方式：
(1) p为 g的左孩子，cur为 p的左孩子时，进行右单旋
(2) 相反，p为 g的右孩子，cur为 p的右孩子时，进行左单旋
(3) p、g变色，p变黑，g变红

• 情况三: cur为红，p为红，g为黑，u不存在/u为黑
  

解决方式：
(1)p为 g的左孩子，cur为 p的右孩子时，则针对 p做左单旋
(2)相反，p为 g的右孩子，cur为 p的右孩子时，则针对 p做右单旋
(3)交换 p和 cur的位置，此时就转换成了情况2来解决

	bool Insert(const 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 (kv.first > cur->_kv.first){parent = cur;cur = cur->_right;}else if (kv.first < cur->_kv.first){parent = cur;cur = cur->_left;}else{return false;}}cur = new Node(kv);cur->_col = RED;if (kv.first > parent->_kv.first){parent->_right = cur;cur->_parent = parent;}else{parent->_left = cur;cur->_parent = parent;}//检测红黑树的性质while (parent && parent->_col == RED){Node* grandfather = parent->_parent;if (parent == grandfather->_left){Node* uncle = grandfather->_right;//情况一：uncle存在且为红if (uncle && uncle->_col == RED){parent->_col = uncle->_col = BLACK;grandfather->_col = RED;//继续向上调整cur = grandfather;parent = cur->_parent;}else //情况二/三：uncle不存在或uncle存在且为黑{//情况三双旋后变单旋，成为情况二if (cur == parent->_right){RotateL(parent);//交换的是节点的指针，不是节点swap(cur, parent);}//处理情况二RotateR(grandfather);grandfather->_col = RED;parent->_col = BLACK;//调整结束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{if (cur == parent->_left){RotateR(parent);swap(cur, parent);}RotateL(grandfather);grandfather->_col = RED;parent->_col = BLACK;break;}}		}_root->_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(){//空树也是红黑树if (nullptr == _root){return true;}//检测根节点是否满足情况if (BLACK != _root->_col){cout << "违反红黑树性质二：根节点必须为黑色" << endl;return false;}//获取任意一条路径中黑色节点的个数size_t blackCount = 0;Node* cur = _root;while (cur){if (BLACK == cur->_col){blackCount++;}			cur = cur->_left;}//检测是否满足红黑树的性质，k用来记录路径中黑色节点的个数size_t k = 0;return _IsValidRBTree(_root, k, blackCount);}bool _IsValidRBTree(Node* root, size_t k, const size_t blackCount){//走到null之后，判断k和blackCount是否相等if (nullptr == root){if (k != blackCount){cout << "违反性质四：每条路径中黑色节点的个数必须相同" << endl;return false;}return true;}//统计黑色节点的个数if (BLACK == root->_col){k++;}//检测当前节点与其双亲是否都为红色Node* parent = root->_parent;if (parent && RED == parent->_col && RED == root->_col){cout << "违反性质三：没有连在一起的红色节点" << endl;return false;}return _IsValidRBTree(root->_left, k, blackCount)&& _IsValidRBTree(root->_right, k, blackCount);}

红黑树的应用

1. C++ STL库 – map/set、mutil_map/mutil_set
2. Java 库
3. linux内核
4. 其他一些库

完整代码实现

#include <iostream>
using namespace std;//节点的颜色
enum Color
{RED,BLACK
};template<class K, class V>
struct RBTreeNode
{RBTreeNode<K, V>* _left;  //左孩子RBTreeNode<K, V>* _right; //右孩子RBTreeNode<K, V>* _parent;//双亲pair<K, V> _kv;Color _col; //红黑颜色限制, 新插入节点默认为红色RBTreeNode(const pair<K, V>& kv, Color col = RED): _left(nullptr), _right(nullptr), _parent(nullptr), _kv(kv), _col(col) {}
};template<class K, class V>
class RBTree
{typedef RBTreeNode<K, V> Node;
public:bool Insert(const 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 (kv.first > cur->_kv.first){parent = cur;cur = cur->_right;}else if (kv.first < cur->_kv.first){parent = cur;cur = cur->_left;}else{return false;}}cur = new Node(kv);cur->_col = RED;if (kv.first > parent->_kv.first){parent->_right = cur;cur->_parent = parent;}else{parent->_left = cur;cur->_parent = parent;}//检测红黑树的性质while (parent && parent->_col == RED){Node* grandfather = parent->_parent;if (parent == grandfather->_left){Node* uncle = grandfather->_right;//情况一：uncle存在且为红if (uncle && uncle->_col == RED){parent->_col = uncle->_col = BLACK;grandfather->_col = RED;//继续向上调整cur = grandfather;parent = cur->_parent;}else //情况二/三：uncle不存在或uncle存在且为黑{//情况三双旋后变单旋，成为情况二if (cur == parent->_right){RotateL(parent);//交换的是节点的指针，不是节点swap(cur, parent);}//处理情况二RotateR(grandfather);grandfather->_col = RED;parent->_col = BLACK;//调整结束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{if (cur == parent->_left){RotateR(parent);swap(cur, parent);}RotateL(grandfather);grandfather->_col = RED;parent->_col = BLACK;break;}}		}_root->_col = BLACK;return true;}Node* Find(const K& key){Node* cur = _root;while (cur){if (key > cur->_kv.first){cur = cur->_right;}else if (key < cur->_kv.first){cur = cur->_left;}else{return cur;}}return nullptr;}void InOrder(){_InOrder(_root);}bool IsValidRBTree(){//空树也是红黑树if (nullptr == _root){return true;}//检测根节点是否满足情况if (BLACK != _root->_col){cout << "违反红黑树性质二：根节点必须为黑色" << endl;return false;}//获取任意一条路径中黑色节点的个数size_t blackCount = 0;Node* cur = _root;while (cur){if (BLACK == cur->_col){blackCount++;}			cur = cur->_left;}//检测是否满足红黑树的性质，k用来记录路径中黑色节点的个数size_t k = 0;return _IsValidRBTree(_root, k, blackCount);}
private:bool _IsValidRBTree(Node* root, size_t k, const size_t blackCount){//走到null之后，判断k和blackCount是否相等if (nullptr == root){if (k != blackCount){cout << "违反性质四：每条路径中黑色节点的个数必须相同" << endl;return false;}return true;}//统计黑色节点的个数if (BLACK == root->_col){k++;}//检测当前节点与其双亲是否都为红色Node* parent = root->_parent;if (parent && RED == parent->_col && RED == root->_col){cout << "违反性质三：没有连在一起的红色节点" << endl;return false;}return _IsValidRBTree(root->_left, k, blackCount)&& _IsValidRBTree(root->_right, k, blackCount);}void _InOrder(Node* root){if (root == nullptr){return;}_InOrder(root->_left);cout << root->_kv.first << ":" << root->_kv.second << endl;_InOrder(root->_right);}//左单旋void RotateL(Node* parent){Node* pParent = parent->_parent;Node* subR = parent->_right;Node* subRL = subR->_left;parent->_right = subRL;if (subRL){subRL->_parent = parent;}subR->_left = parent;parent->_parent = subR;if (_root == parent){_root = subR;subR->_parent = nullptr;}else{if (parent == pParent->_left){pParent->_left = subR;}else{pParent->_right = subR;}subR->_parent = pParent;}}//右单旋void RotateR(Node* parent){Node* pParent = parent->_parent;Node* subL = parent->_left;Node* subLR = subL->_right;parent->_left = subLR;if (subLR){subLR->_parent = parent;}subL->_right = parent;parent->_parent = subL;if (_root == parent){_root = subL;subL->_parent = nullptr;}else{if (parent == pParent->_left){pParent->_left = subL;}else{pParent->_right = subL;}subL->_parent = pParent;}}
private:Node* _root = nullptr;
};

测试用例

void Test()
{int arr[] = { 16, 3, 7, 11, 9, 26, 18, 14, 15 };RBTree<int, int> t;for (auto e : arr){t.Insert(make_pair(e, e));}t.InOrder();cout << t.IsValidRBTree() << endl;RBTreeNode<int, int>* ret = t.Find(26);if (ret){cout << "找到了:" << ret->_kv.first << endl;}else{cout << "没找到" << endl;}
}