目录

红黑树的概念

红黑树的节点结构定义

红黑树的插入

红黑树的验证

实现红黑树完整代码

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红黑树的概念

有了上述限制，最短路径就是只包含黑色节点的路径，而最长路径就是黑红交替，所以所有路径长度范围都在[最短路径，最短路径的2倍]这个区间之间， 保证了红黑树近似平衡

红黑树的节点结构定义

//颜色定义成枚举类型
enum Color
{RED,BLACK
};//节点结构定义
template<class K, class V>
struct RBTreeNode
{//三叉链RBTreeNode<K, V>* _left;RBTreeNode<K, V>* _right;RBTreeNode<K, V>* _parent;//键值对pair<K, V> _kv;//颜色变量Color _col;RBTreeNode(const pair<K, V>& kv):_left(nullptr), _right(nullptr), _parent(nullptr), _kv(kv), _col(RED){}
};

红黑树的插入

插入新节点我们选择插入红色节点，理由如下:

1.插入黑色节点，一定破坏性质4，所有路径的黑色节点个数都要变化

2.插入红色节点，可能破坏性质3，插入红色节点的父亲是黑色节点，没有破坏性质3, 不需要调整；插入红色节点的父亲是红色节点，破坏了性质3， 需要进行调整

综合考虑，插入红色节点的代价相比插入黑色节点小很多，因此我们选择插入红色节点

插入情况的分类:

上面已经提到，插入红色节点的父亲是黑色节点，没有破坏性质3，因此无需调整，因此下面只讨论插入红色节点的父亲是红色节点的情况

处理方法总体分两类

1.插入节点的叔叔节点存在且为红色 --- 变色即可

字母含义: cur --- 插入节点， p --- cur的父亲节点,  g --- cur的爷爷节点, u --- cur的叔叔节点

ps: g一定存在，因为p是红色节点, 不可能为根节点, 而g也一定是黑色节点，因为如果是红色节点，说明在cur插入之前红黑树就已经出问题了！

插入之后cur和p都是红色，因为不能出现连续红色节点，我们选择把p变黑，而p变黑之后，g-p-cur的这条路径多了一个黑色节点，因此我们又把g变红，g变红之后，g-u的这条路径又少了一个黑色节点，由于叔叔节点存在且为红色，因此我们又把u变黑即可

ps:变色完之后，发现g变成了红色，而上图的树可能只是一颗子树，因此g变红之后，接着:

1.如果g是根，把g变黑，因为红黑树要求根节点是黑色

2.如果g不是根，g必然有父亲，继续判断g的父亲节点的颜色，如果g的父节点是黑色，停止处理（因为没有再违反红黑树规则了）；如果g的父节点是红色，此时又出现了连续的红节点，需要继续处理, 把cur更新到g的位置，继续循环处理!

2.插入节点的叔叔节不存在 或者 存在且为黑色 --- 旋转 + 变色

这种情况下，单纯的变色已经解决不了问题了！需要 旋转+变色 来处理

具体采用哪种旋转方法，在前面的博客 实现AVL树-CSDN博客 已经讲解过了

注意:无论是哪种旋转+变色方式，发现最终当前子树的根节点都变成了黑色，此时无论cur的父节点是黑色还是红色都不违反红黑树的规则，因此旋转+变色完成之后，直接break跳出循环即可！

右单旋 + 变色 (p是g的左，cur是p的左)

左右双旋 + 变色 (p是g的左，cur是p的右)

 左单旋 + 变色 (p是g的右，cur是p的右)

右左双旋 + 变色 (p是g的右，cur是p的左)

红黑树旋转代码(除了不需要调平衡因子，其他代码与AVL树一样)：

	void RotateL(Node* parent){Node* subR = parent->_right;Node* subRL = subR->_left;parent->_right = subRL;subR->_left = parent;Node* parentParent = parent->_parent;parent->_parent = subR;if (subRL)subRL->_parent = parent;if (parentParent == nullptr){_root = subR;_root->_parent = nullptr;}else{if (parentParent->_left == parent){parentParent->_left = subR;}else{parentParent->_right = subR;}subR->_parent = parentParent;}}//右单旋void RotateR(Node* parent){Node* subL = parent->_left;Node* subLR = subL->_right;parent->_left = subLR;if (subLR)subLR->_parent = parent;Node* parentParent = parent->_parent;subL->_right = parent;parent->_parent = subL;if (parentParent == nullptr){_root = subL;_root->_parent = nullptr;}else{if (parentParent->_left == parent){parentParent->_left = subL;}else{parentParent->_right = subL;}subL->_parent = parentParent;}}//左右双旋void RotateLR(Node* parent){RotateL(parent->_left);RotateR(parent);}//右左双旋void RotateRL(Node* parent){RotateR(parent->_right);RotateL(parent);}

红黑树插入代码：

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 (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);if (parent->_kv.first < kv.first){parent->_right = cur;cur->_parent = parent;}else{parent->_left = cur;cur->_parent = parent;}while (parent && parent->_col == RED) //父亲不存在或者父亲为黑色就不需要继续调整了!{Node* grandfather = parent->_parent;if (grandfather->_left == parent) //父亲是爷爷的左孩子{Node* uncle = grandfather->_right;//1.插入节点的叔叔存在且为红色 --- 父亲和叔叔变黑，爷爷变红if (uncle && uncle->_col == RED){parent->_col = uncle->_col = BLACK;grandfather->_col = RED;//调整完之后，更新cur, 判断是否还需要调整cur = grandfather;parent = cur->_parent;}//2.uncle不存在 / uncle为黑色 --- 旋转 + 变色else{if (cur == parent->_left) //单纯左边高{RotateR(grandfather); //右单旋//变色parent->_col = BLACK;grandfather->_col = RED;}else //cur是parent的右边{RotateLR(grandfather); //左右双旋//变色cur->_col = BLACK;grandfather->_col = RED;}break; //旋转+变色完之后, 该子树的根变成了黑色，无需继续调整}}else //父亲是爷爷的右孩子{Node* uncle = grandfather->_left;//1.插入节点的叔叔存在且为红色 --- 父亲和叔叔变黑，爷爷变红if (uncle && uncle->_col == RED){parent->_col = uncle->_col = BLACK;grandfather->_col = RED;//调整完之后，更新cur, 判断是否还需要调整cur = grandfather;parent = cur->_parent;}//2.uncle不存在 / uncle为黑色 --- 旋转 + 变色else{if (cur == parent->_left) //cur是parent的左边，进行右单旋{RotateRL(grandfather); //右左单旋//变色cur->_col = BLACK;grandfather->_col = RED;}else //cur是parent的右边{RotateL(grandfather); //左单旋//变色grandfather->_col = RED;parent->_col = BLACK;}break; //旋转+变色完之后, 该子树的根变成了黑色，无需继续调整}}}_root->_col = BLACK;return true;
}

红黑树的验证

判断是否是红黑树

	bool IsRBTree(){if (_root == nullptr) return true;if (_root->_col == RED){cout << "根节点为红色" << endl;return false;}//以最左路径的黑节点的个数作为参考值Node* cur = _root;int BlackCount = 0;while (cur){if (cur->_col == BLACK)BlackCount++;cur = cur->_left;}//调用子函数判断红黑树int count = 0;return _IsRBTree(_root, count, BlackCount);}
private:bool _IsRBTree(Node* root, int count, int BlackCount){if (root == nullptr){if (count != BlackCount) {cout << "黑色节点的个数不相等" << endl;return false;}return true;}//判断节点的孩子节点颜色不好判断, 转化成判断父亲节点颜色if (root->_col == RED && root->_parent->_col == RED){cout << "存在连续的红节点" << endl;return false;}if (root->_col == BLACK)count++;//递归子问题return _IsRBTree(root->_left, count, BlackCount)&& _IsRBTree(root->_right, count, BlackCount);}

验证红黑树

#include "RBTree.h"#include <vector>
void test1()
{int a[] = { 16, 3, 7, 11, 9, 26, 18, 14, 15 };RBTree<int, int> t;for (auto e : a){t.Insert(make_pair(e, e));}t.IsRBTree();cout << t.IsRBTree() << endl;
}void test2()
{const int N = 30;vector<int> v;v.reserve(N);srand(time(0));for (size_t i = 0; i < N; i++){v.push_back(rand());cout << v.back() << endl;}RBTree<int, int> t;for (auto e : v){t.Insert(make_pair(e, e));cout << "Insert:" << e << "->" << t.IsRBTree() << endl;}cout << t.IsRBTree() << endl;
}int main()
{//test1();test2();return 0;
}

实现红黑树完整代码

#pragma once#include <iostream>
using namespace std;//颜色定义成枚举类型
enum Color
{RED,BLACK
};//节点结构定义
template<class K, class V>
struct RBTreeNode
{//三叉链RBTreeNode<K, V>* _left;RBTreeNode<K, V>* _right;RBTreeNode<K, V>* _parent;//键值对pair<K, V> _kv;//颜色变量Color _col;RBTreeNode(const 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://红黑树中选择插入红色节点//插入黑色节点后所有路径的黑色节点数量都要变化，很麻烦~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 (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);if (parent->_kv.first < kv.first){parent->_right = cur;cur->_parent = parent;}else{parent->_left = cur;cur->_parent = parent;}while (parent && parent->_col == RED) //父亲不存在或者父亲为黑色就不需要继续调整了!{Node* grandfather = parent->_parent;if (grandfather->_left == parent) //父亲是爷爷的左孩子{Node* uncle = grandfather->_right;//1.插入节点的叔叔存在且为红色 --- 父亲和叔叔变黑，爷爷变红if (uncle && uncle->_col == RED){parent->_col = uncle->_col = BLACK;grandfather->_col = RED;//调整完之后，更新cur, 判断是否还需要调整cur = grandfather;parent = cur->_parent;}//2.uncle不存在 / uncle为黑色 --- 旋转 + 变色else{if (cur == parent->_left) //单纯左边高{RotateR(grandfather); //右单旋//变色parent->_col = BLACK;grandfather->_col = RED;}else //cur是parent的右边{RotateLR(grandfather); //左右双旋//变色cur->_col = BLACK;grandfather->_col = RED;}break; //旋转+变色完之后, 该子树的根变成了黑色，无需继续调整}}else //父亲是爷爷的右孩子{Node* uncle = grandfather->_left;//1.插入节点的叔叔存在且为红色 --- 父亲和叔叔变黑，爷爷变红if (uncle && uncle->_col == RED){parent->_col = uncle->_col = BLACK;grandfather->_col = RED;//调整完之后，更新cur, 判断是否还需要调整cur = grandfather;parent = cur->_parent;}//2.uncle不存在 / uncle为黑色 --- 旋转 + 变色else{if (cur == parent->_left) //cur是parent的左边，进行右单旋{RotateRL(grandfather); //右左单旋//变色cur->_col = BLACK;grandfather->_col = RED;}else //cur是parent的右边{RotateL(grandfather); //左单旋//变色grandfather->_col = RED;parent->_col = BLACK;}break; //旋转+变色完之后, 该子树的根变成了黑色，无需继续调整}}}_root->_col = BLACK;return true;}//左单旋void RotateL(Node* parent){Node* subR = parent->_right;Node* subRL = subR->_left;parent->_right = subRL;subR->_left = parent;Node* parentParent = parent->_parent;parent->_parent = subR;if (subRL)subRL->_parent = parent;if (parentParent == nullptr){_root = subR;_root->_parent = nullptr;}else{if (parentParent->_left == parent){parentParent->_left = subR;}else{parentParent->_right = subR;}subR->_parent = parentParent;}}//右单旋void RotateR(Node* parent){Node* subL = parent->_left;Node* subLR = subL->_right;parent->_left = subLR;if (subLR)subLR->_parent = parent;Node* parentParent = parent->_parent;subL->_right = parent;parent->_parent = subL;if (parentParent == nullptr){_root = subL;_root->_parent = nullptr;}else{if (parentParent->_left == parent){parentParent->_left = subL;}else{parentParent->_right = subL;}subL->_parent = parentParent;}}//左右双旋void RotateLR(Node* parent){RotateL(parent->_left);RotateR(parent);}//右左双旋void RotateRL(Node* parent){RotateR(parent->_right);RotateL(parent);}//判断是否是红黑树bool IsRBTree(){if (_root == nullptr) return true;if (_root->_col == RED){cout << "根节点为红色" << endl;return false;}//以最左路径的黑节点的个数作为参考值Node* cur = _root;int BlackCount = 0;while (cur){if (cur->_col == BLACK)BlackCount++;cur = cur->_left;}//调用子函数判断红黑树int count = 0;return _IsRBTree(_root, count, BlackCount);}
private:bool _IsRBTree(Node* root, int count, int BlackCount){if (root == nullptr){if (count != BlackCount) {cout << "黑色节点的个数不相等" << endl;return false;}return true;}//判断节点的孩子节点颜色不好判断, 转化成判断父亲节点颜色if (root->_col == RED && root->_parent->_col == RED){cout << "存在连续的红节点" << endl;return false;}if (root->_col == BLACK)count++;//递归子问题return _IsRBTree(root->_left, count, BlackCount)&& _IsRBTree(root->_right, count, BlackCount);}private:Node* _root = nullptr;
};