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目录

一、红黑树的概念

​编辑二、红黑树的性质

三、红黑树节点的定义

四、红黑树结构

五、红黑树的插入操作

5.1. 按照二叉搜索的树规则插入新节点

5.2、检测新节点插入后，红黑树的性质是否造到破坏

 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为黑

情况三: cur为红，p为红，g为黑，u不存在/u为黑

六、红黑树的验证

七、红黑树与AVL树的比较

八、key结构红黑树整体代码

九、key，value 结构红黑树整体代码

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一、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俩倍，因而是接近平衡的。

二、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)

三、红黑树节点的定义

红黑树的节点，我们这里使用的是三叉链，方便对后面内容的操作

首先，我们定义了一个枚举常量，来表示红黑树节点的颜色

其次，定义节点，一个红黑树的节点包含<左孩子，右孩子，父节点，数据，颜色>

接着我们定义构造函数<对其节点数据进行初始化>，左右孩子和父节点置空，插入的颜色默认为红色，数据为传入的数据。

enum COLOR
{BLACK,RED
};
template <class T>
struct RBNode
{RBNode<T>* _left;RBNode<T>* _right;RBNode<T>* _parent;T _value;COLOR _color;//颜色RBNode(const T & value=T()):_left(nullptr), _right(nullptr), _parent(nullptr), _value(value), _color(RED){}
};

四、红黑树结构

为了后续封装map和set简单一些，在红黑树的实现中增加一个头结点，因为根节点必须为黑色，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 pParent 域指向红黑树的根节点，pLeft域指向红黑树中最小的节点，_pRight域指向红黑树中最大的节点，如下：

五、红黑树的插入操作

相比于AVL树，插入比较简单，效率比较高，红黑树比AVL树的调整次数要少。

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（有破坏进行调整）      

5.1. 按照二叉搜索的树规则插入新节点

bool Insert(const T& value){// 1. 按照二叉搜索的树方式插入新节点//搜索树的插入if (_header->_parent == nullptr){//空树，创建根节点pNode root = new Node(value);root->_color = BLACK;root->_parent = _header;_header->_parent = root;_header->_left = root;_header->_right = root;return true;}//从根开始搜索pNode cur = _header->_parent;pNode parent = nullptr;//查找插入的位置while (cur){parent = cur;//按照key值确定位置, key不能重复if (cur->_value == value)return false;else if (cur->_value > value)cur = cur->_left;elsecur = cur->_right;}//节点创建cur = new Node(value);//节点插入if (parent->_value > cur->_value)parent->_left = cur;elseparent->_right = cur;//节点连接cur->_parent = parent;//while()// {// 2. 检测新节点插入后，红黑树的性质是否造到破坏，// 若满足直接退出，否则对红黑树进行旋转着色处理//}// 根节点的颜色可能被修改，将其改回黑色_header->_parent->_color = BLACK;//更新 _header->_left, _header->_right_header->_left = leftMost();_header->_right = rightMost();return true;}

5.2、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

       注意::此时看到的树，又可能是一颗完整的树，也有可能是一颗子树

      如果g是根节点，调整完成后，需要把根节点改为黑色

      如果g是子树，g一定有双亲（父亲和叔叔），如果g的双亲为红色，则继续往上调整

解决方式：将p，u改为黑，g改为红，然后把g当作cur，继续往上调整，直至cur为根节点（根为黑）

情况二: cur为红，p为红，g为黑，u不存在/u为黑

说明：

U有俩种情况：

1.如果U节点不存在，则cur一定是新插入的节点，因为如果cur不为新插入的节点，则cur和p一定有一个节点的颜色为黑色，就不满足性质4：每条路径黑色节点相同

2.如果U节点存在，则其一定是黑色的，那么cue节点原来的颜色一定是黑色的，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为红色。

解决方式：p为g的左孩子，cur为p的左孩子，则进行右单旋转；相反，
                  p为g的右孩子，cur为p的右孩子，则进行左单旋转
                  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：p为g的左孩子，cur为p的右孩子，则针对p做左单旋转；相反，
                  p为g的右孩子，cur为p的左孩子，则针对p做右单旋转
               则转换成了情况2

bool insert(const T& value){//搜索树的插入if (_header->_parent == nullptr){//空树，创建根节点pNode root = new Node(value);root->_color = BLACK;root->_parent = _header;_header->_parent = root;_header->_left = root;_header->_right = root;return true;}//从根开始搜索pNode cur = _header->_parent;pNode parent = nullptr;//查找插入的位置while (cur){parent = cur;//按照key值确定位置, key不能重复if (cur->_value == value)return false;else if (cur->_value > value)cur = cur->_left;elsecur = cur->_right;}//节点创建cur = new Node(value);//节点插入if (parent->_value > cur->_value)parent->_left = cur;elseparent->_right = cur;//节点连接cur->_parent = parent;//调整和更新（颜色）：连续红色需要调整while (cur != _header->_parent && cur->_parent->_color == RED)//当前不是根，并且你的父亲是红色{//cur:当前节点，parent:父亲节点， gfather：祖父节点，uncle:叔叔节点parent = cur->_parent;pNode gfather = parent->_parent;if (gfather->_left == parent){pNode uncle = gfather->_right;//uncle 存在且为红if (uncle && uncle->_color == RED){//修改颜色parent->_color = uncle->_color = BLACK;gfather->_color = RED;//继续向上更新cur = gfather;}else{//如果存在双旋的场景，可以先进行一次单旋，使它变成单旋的场景if (cur == parent->_right){RotateL(parent);swap(cur, parent);}//右旋RotateR(gfather);//修改颜色parent->_color = BLACK;gfather->_color = RED;//停止调整break;}}//gfather->_right == parentelse{pNode uncle = gfather->_left;if (uncle && uncle->_color == RED){//修改颜色uncle->_color = parent->_color = BLACK;gfather->_color = RED;cur = gfather;}else{//判断是否有双旋的场景if (cur == parent->_left){//以parent右旋RotateR(parent);//交换指针swap(cur, parent);}//以gfather 左旋RotateL(gfather);//修改颜色parent->_color = BLACK;gfather->_color = RED;//停止调整break;}}}//根的颜色始终是黑的 根：_header->_parent_header->_parent->_color = BLACK;//更新 _header->_left, _header->_right_header->_left = leftMost();_header->_right = rightMost();return true;}

六、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 isRBTree(){pNode root = _header->_parent;if (root == nullptr)return true;if (root->_color == RED){cout << "根节点必须是黑色的!!!" << endl;return false;}//根节点是黑色//需要判断每条路径上黑色个数相同//可以先任意遍历一条路径 比如走最右路径。查找black数量pNode cur = root;int blackCount = 0;while (cur){if (cur->_color == BLACK)++blackCount;cur = cur->_right;}int k = 0;return _isRBTree(root, k, blackCount);}bool  _isRBTree(pNode root, int curBlackCount, int totalBlackCout)//curBlackCount:走到当前节点黑色个数{//每条路径上黑色个数相同//没有连续红色结点//一条路径走完if (root == nullptr){if (curBlackCount != totalBlackCout){cout << "每条路径-黑色结点个数不同" << endl;return false;}return true;}if (root->_color == BLACK)++curBlackCount;//没有红色连续pNode parent = root->_parent;if (parent->_color == RED && root->_color == RED){cout << "有连续的红色结点" << endl;return false;}return  _isRBTree(root->_left, curBlackCount, totalBlackCout) && _isRBTree(root->_right, curBlackCount, totalBlackCout);}

七、红黑树与AVL树的比较

红黑树和AVL树都是高效的平衡二叉树，增删改查的时间复杂度都是O( )，红黑树不追求绝对平衡，其只需保证最长路径不超过最短路径的2倍，相对而言，降低了插入和旋转的次数，所以在经常进行增删的结构中性能比AVL树更优，而且红黑树实现比较简单，所以实际运用中红黑树更多。
 

八、key结构红黑树整体代码

#define _CRT_SECURE_NO_WARNINGS 1
#include<time.h>
#include<utility>
#include<iostream>
using namespace std;
enum COLOR
{BLACK,RED
};
template <class T>
struct RBNode
{RBNode<T>* _left;RBNode<T>* _right;RBNode<T>* _parent;T _value;COLOR _color;//颜色RBNode(const T & value=T()):_left(nullptr), _right(nullptr), _parent(nullptr), _value(value), _color(RED){}
};template <class T>
class RBTree
{
public:typedef RBNode<T> Node;typedef Node* pNode;RBTree(){//构建空的红黑树  空树--》带头的红黑树，头不是根_header = new Node;_header->_left = _header;_header->_right = _header;}/*红黑树插入：1.相对于AVL树，插入比较简单，且效率高，红黑树比AVL树调整次数要少2.二叉树进行插入3.判断有没有连续的红色结点如果有：a:只需要修改颜色： uncle为红色b:修改颜色，旋转：u不存在、存在且为黑单旋：cur,parent在gfather的同一边双旋：cur,parent不在gfather的同一边，首先经过一次单璇，交换指针，转化为上面单璇场景，没有：不需要做任何操作，插入结束。*/bool insert(const T& value){//搜索树的插入if (_header->_parent == nullptr){//空树，创建根节点pNode root = new Node(value);root->_color = BLACK;root->_parent = _header;_header->_parent = root;_header->_left = root;_header->_right = root;return true;}//从根开始搜索pNode cur = _header->_parent;pNode parent = nullptr;//查找插入的位置while (cur){parent = cur;//按照key值确定位置, key不能重复if (cur->_value == value)return false;else if (cur->_value > value)cur = cur->_left;elsecur = cur->_right;}//节点创建cur = new Node(value);//节点插入if (parent->_value > cur->_value)parent->_left = cur;elseparent->_right = cur;//节点连接cur->_parent = parent;//调整和更新（颜色）：连续红色需要调整while (cur != _header->_parent && cur->_parent->_color == RED)//当前不是根，并且你的父亲是红色{//cur:当前节点，parent:父亲节点， gfather：祖父节点，uncle:叔叔节点parent = cur->_parent;pNode gfather = parent->_parent;if (gfather->_left == parent){pNode uncle = gfather->_right;//uncle 存在且为红if (uncle && uncle->_color == RED){//修改颜色parent->_color = uncle->_color = BLACK;gfather->_color = RED;//继续向上更新cur = gfather;}else{//如果存在双旋的场景，可以先进行一次单旋，使它变成单旋的场景if (cur == parent->_right){RotateL(parent);swap(cur, parent);}//右旋RotateR(gfather);//修改颜色parent->_color = BLACK;gfather->_color = RED;//停止调整break;}}//gfather->_right == parentelse{pNode uncle = gfather->_left;if (uncle && uncle->_color == RED){//修改颜色uncle->_color = parent->_color = BLACK;gfather->_color = RED;cur = gfather;}else{//判断是否有双旋的场景if (cur == parent->_left){//以parent右旋RotateR(parent);//交换指针swap(cur, parent);}//以gfather 左旋RotateL(gfather);//修改颜色parent->_color = BLACK;gfather->_color = RED;//停止调整break;}}}//根的颜色始终是黑的 根：_header->_parent_header->_parent->_color = BLACK;//更新 _header->_left, _header->_right_header->_left = leftMost();_header->_right = rightMost();return true;}pNode leftMost(){pNode cur = _header->_parent;while (cur && cur->_left != nullptr){cur = cur->_left;}return cur;}pNode rightMost(){pNode cur = _header->_parent;while (cur && cur->_right != nullptr){cur = cur->_right;}return cur;}void RotateR(pNode parent){pNode subL = parent->_left;pNode subLR = subL->_right;// 1subL->_right = parent;// 2parent->_left = subLR;// 3if (subLR)subLR->_parent = parent;// 4,  5if (parent != _header->_parent){// subL <---> parent->parentpNode gParent = parent->_parent;if (gParent->_left == parent)gParent->_left = subL;elsegParent->_right = subL;subL->_parent = gParent;}else{//更新根节点_header->_parent = subL;//subL->_parent = nullptr;subL->_parent = _header;}// 6parent->_parent = subL;}void RotateL(pNode parent){pNode subR = parent->_right;pNode subRL = subR->_left;subR->_left = parent;parent->_right = subRL;if (subRL)subRL->_parent = parent;if (parent != _header->_parent) {pNode gParent = parent->_parent;if (gParent->_left == parent)gParent->_left = subR;elsegParent->_right = subR;subR->_parent = gParent;}else{_header->_parent = subR;//根的父节点不是nullptr//subR->_parent = nullptr;subR->_parent = _header;}parent->_parent = subR;}void inOrder(){_inOrder(_header->_parent);}void _inOrder(pNode root){if (root) {_inOrder(root->_left);cout << root->_value<<endl;_inOrder(root->_right);}}bool isRBTree(){pNode root = _header->_parent;if (root == nullptr)return true;if (root->_color == RED){cout << "根节点必须是黑色的!!!" << endl;return false;}//根节点是黑色//需要判断每条路径上黑色个数相同//可以先任意遍历一条路径 比如走最右路径。查找black数量pNode cur = root;int blackCount = 0;while (cur){if (cur->_color == BLACK)++blackCount;cur = cur->_right;}int k = 0;return _isRBTree(root, k, blackCount);}bool  _isRBTree(pNode root, int curBlackCount, int totalBlackCout)//curBlackCount:走到当前节点黑色个数{//每条路径上黑色个数相同//没有连续红色结点//一条路径走完if (root == nullptr){if (curBlackCount != totalBlackCout){cout << "每条路径-黑色结点个数不同" << endl;return false;}return true;}if (root->_color == BLACK)++curBlackCount;//没有红色连续pNode parent = root->_parent;if (parent->_color == RED && root->_color == RED){cout << "有连续的红色结点" << endl;return false;}return  _isRBTree(root->_left, curBlackCount, totalBlackCout) && _isRBTree(root->_right, curBlackCount, totalBlackCout);}private:pNode _header;
};

九、key，value 结构红黑树整体代码

#pragma once
#include<iostream>
using namespace std;//定义颜色
enum Color
{RED,BLACK,
};// 定义节点
template<class K, class V>
struct RBTreeNode
{pair<K, V> _kv;RBTreeNode<K, V>* _left;RBTreeNode<K, V>* _right;RBTreeNode<K, V>* _parent;Color _col;RBTreeNode(const pair<K, V>& kv):_kv(kv), _left(nullptr), _right(nullptr), _parent(nullptr), _col(RED){}
};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;}else{Node* cur = _root;Node* parent = nullptr;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;}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;cur = grandfather;parent = cur->_parent;}else{// 情况二if (cur == parent->_left){RotateR(grandfather);parent->_col = BLACK;grandfather->_col = RED;}// 情况三else{RotateL(parent);RotateR(grandfather);cur->_col = BLACK;grandfather->_col = RED;}break;}}else{Node* uncle = grandfather->_left;if (uncle && uncle->_col == RED){uncle->_col = parent->_col = BLACK;grandfather->_col = RED;cur = grandfather;parent = cur->_parent;}else{// g//     p// cif (cur == parent->_left){RotateR(parent);RotateL(grandfather);cur->_col = BLACK;grandfather->_col = RED;}// g//    p//      celse{RotateL(grandfather);parent->_col = BLACK;grandfather->_col = RED;}break;}}}_root->_col = BLACK;return true;}}void RotateL(Node* parent){Node* SubR = parent->_right;Node* SubRL = SubR->_left;parent->_right = SubRL;if (SubRL)SubRL->_parent = parent;Node* ppNode = parent->_parent;SubR->_left = parent;parent->_parent = SubR;if (ppNode == nullptr){_root = SubR;SubR->_parent = nullptr;}else{if (parent == ppNode->_left){ppNode->_left = SubR;SubR->_parent = ppNode;}else{ppNode->_right = SubR;SubR->_parent = ppNode;}}}void RotateR(Node* parent){Node* SubL = parent->_left;Node* SubLR = SubL->_right;parent->_left = SubLR;if (SubLR)SubLR->_parent = parent;Node* ppNode = parent->_parent;SubL->_right = parent;parent->_parent = SubL;if (ppNode == nullptr){_root = SubL;SubL->_parent = nullptr;}else{if (parent == ppNode->_left){ppNode->_left = SubL;}else{ppNode->_right = SubL;}SubL->_parent = ppNode;}}void InOrder(){_InOrder(_root);}void _InOrder(Node* root){if (root == nullptr)return;_InOrder(root->_left);cout << root->_kv.first << ":" << root->_kv.second << endl;_InOrder(root->_right);}bool check(Node* root, int blackNum, int ref){if (root == nullptr){if (blackNum != ref){cout << "违反规则:本条路径的黑色节点的数量跟最左路径不相等" << endl;return false;}return true;}if (root->_col == RED && root->_parent->_col == RED){cout << "违反规则：出现连续红色节点" << endl;return false;}if (root->_col == BLACK)blackNum++;return check(root->_left, blackNum, ref)&& check(root->_right, blackNum, ref);}bool IsBalance(){if (_root == nullptr)return true;if (_root->_col != BLACK)return false;int ref = 0;// 统计黑节点的个数Node* left = _root;while (left){if (left->_col == BLACK)ref++;left = left->_left;}return check(_root, 0, ref);}
private:Node* _root = nullptr;
};