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【C++】红黑树的全面探索和深度解析

news 来源：原创 2024/9/20 9:40:29

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1. 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

📙红黑树由Rudolf Bayer在1972年发明，最初被称为平衡二叉B树（Symmetric Binary B-trees），后来被Guibas和Robert Sedgewick修改为如今的“红黑树”。

2. 红黑树的性质

节点是红色或黑色：每个节点都有一个颜色属性，颜色可以是红色或黑色。

根节点是黑色：树的根节点必须是黑色。

红色节点的子节点是黑色：如果一个节点是红色，则它的两个子节点必须是黑色（即不能有两个连续的红色节点）。

每个节点到其每个叶子节点的路径都包含相同数量的黑色节点：从任何节点到其每个叶子节点的路径上，经过的黑色节点的数量必须相同。

叶子节点是黑色：红黑树的叶子节点（通常是指空节点）被视为黑色。

3. 红黑树的节点结构及定义

红黑树节点的定义通常包含以下几个关键部分：

🧩3.1 基本元素

_left：指向节点的左子节点的指针
_right：指向节点的右子节点的指针
_parent：指向节点的父节点的指针
_kv：一个结构体或配对（pair），包含节点的键值（key）和值（value）。这取决于红黑的具体用途，可能只包含键或包含键值对。
_col：表示当前节点的颜色。

🧩3.2 节点颜色（_col）

在上面的定义中，_col 成员变量用于表示节点的颜色，通过 Color 枚举类型来定义，可以是 RED 或 BLACK。

🧩3.3 构造函数

初始化一个新节点时，通常需要一个构造函数，它接受一个键值对（或仅键），并设置节点的左子节点、右子节点、父节点和颜色（初始化为红色）

🧩3.4 BR节点定义：

template<class K, class V>
struct BSTreeNode
{BSTreeNode<K, V>* _left;    //左子树BSTreeNode<K, V>* _right;   //右子树BSTreeNode<K, V>* _parent;  //父亲pair<K, V> _kv;       //存放节点值的string _col;    //颜色（通过这个可以直到左右子树存在情况）//构造函数BSTreeNode(const pair<K, V>& kv):_left(nullptr), _right(nullptr), _parent(nullptr), _kv(kv), _col("RED")     //默认颜色为红色{}
};

红黑树的节点结构与二叉搜索树和AVL树差别不大，最大的差别就是加入了一个新的存储点——颜色

4. 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：

按照二叉搜索的树规则插入新节点
检测新节点插入后，红黑树的性质是否造到破坏

在我们进行插入操作之前，我们先定义一个红黑树的类

红黑树定义：

template<class K, class V>
class RBTTree
{typedef BSTreeNode<K, V> Node;
public:// 其他未实现的成员函数
private:Node* _root = nullptr;	
};

红黑树的插入操作类似于我们之前AVL树的插入，只不过红黑树的插入操作涉及到旋转操作以及考虑其他节点的颜色，前面的操作还是一样的

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){parent = cur;if (cur->_kv.first < kv.first){cur = cur->_right;}else if (cur->_kv.first > kv.first){cur = cur->_left;}else{return false;}}// 新增节点给红色cur = new Node(kv);cur->_col = RED;if (parent->_kv.first < kv.first){parent->_right = cur;}else{parent->_left = cur;}cur->_parent = parent;// 检测新节点插入后，红黑树的性质是否造到破坏return true;}

🧩检测红黑树是否造到破坏

（如果遭到破坏则对当前红黑树进行变色，旋转处理）
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

解决方式：将parent,uncle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

解决方式： p为g的左孩子，cur为p的右孩子，则针对p做左右双旋；p为g的右孩子，cur为p的左孩子，则针对p做右左旋转

cur、grandfather变色–> c变黑，p变红

检测红黑树是否造到破坏代码演示(C++)：

while (parent && parent->_col == RED) //当父亲节点为红色，则出现了连续的红色，不符合条件{Node* grandfather = parent->_parent;//    g//  p   uif (parent == grandfather->_left) {	Node* uncle = grandfather->_right;if (uncle && uncle->_col == RED) //叔叔存在并且为红{parent->_col = uncle->_col = BLACK;grandfather->_col = RED;cur = grandfather;parent = cur->_parent; //往上面走}else{//u存在且为黑或不存在 ->变色再继续往上处理 + 变色if (cur == parent->_left) { //cur存在那么cur一定为红色 //    g//  p   u//c//单旋，把p旋转上去，p作为子树根节点，g作为p的右RotateR(grandfather);parent->_col = BLACK;grandfather->_col = RED;}else{//    g//  p   u//    c//双旋，将cur旋转上去，p作为cur的左，然后再旋转把cur旋转上去，g作为cur右边RotateL(parent);RotateR(grandfather);cur->_col = BLACK;grandfather->_col = RED;}break;}}else{//    g//  u   pNode* 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;}}}

红黑树的旋转和AVL树差不多，我们直接上代码回顾以下：
旋转代码示例(C++)：

void RotateL(Node* parent) // 左旋
{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;// 判断parent是不是根节点if (_root == parent){_root = subR;subR->_parent = nullptr;}else{if (parent == Parentparent->_left)Parentparent->_left = subR;elseParentparent->_right = subR;subR->_parent = Parentparent;}
}void RotateR(Node* parent) // 右旋
{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(_root == parent){_root = subL;subL->_parent = nullptr;}else{if (parent == Parentparent->_left)Parentparent->_left = subL;elseParentparent->_right = subL;subL->_parent = Parentparent;}
}

5. 红黑树的验证

📝红黑树的检测分为两步：

检测其是否满足二叉搜索树(中序遍历是否为有序序列)
检测其是否满足红黑树的性质

中序遍历代码演示(C++)：

void InOrder()
{_InOrder(_root);cout << endl;
}
void _InOrder(Node* root)
{if (root == nullptr) return;_InOrder(root->_left);_cout << root->_kv.first << ":" << root->_kv.second << endl;_InOrder(root->_right);
}

检测其是否满足红黑树的性质(C++)：

bool IsBalance()
{if (_root == nullptr)return true;if (_root->_col == RED){return false;}// 随便找条路径作为参考值int refNum = 0;Node* cur = _root;while (cur){if (cur->_col == BLACK){++refNum;}cur = cur->_left;}return Check(_root, 0, refNum);
}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);
}

测试代码用例及结果：

6. 红黑树的完整代码及总结

#include<iostream>
#include<vector>
#include<assert.h>
using namespace std;enum Colour
{RED,BLACK
};template<class K, class V>
struct RBTreeNode
{pair<K, V> _kv;RBTreeNode<K, V>* _left;RBTreeNode<K, V>* _right;RBTreeNode<K, V>* _parent;Colour _col;RBTreeNode(const pair<K, V>& kv):_kv(kv), _left(nullptr), _right(nullptr), _parent(nullptr){}
};template<class K, class V>
class RBTree
{typedef RBTreeNode<K, V> Node;
public:RBTree() = default;RBTree(const RBTree<K, V>& t){_root = Copy(t._root);}RBTree<K, V>& operator=(RBTree<K, V> t){swap(_root, t._root);return *this;}~RBTree(){Destroy(_root);_root = nullptr;}bool Insert(const pair<K, V>& kv){if (_root == nullptr){_root = new Node(kv);_root->_col = BLACK; //根节点默认为黑色return true;}Node* parent = nullptr;Node* cur = _root;while (cur){parent = cur;if (cur->_kv.first < kv.first) cur = cur->_right;else if (cur->_kv.first > kv.first) cur = cur->_left;else return false;}cur = new Node(kv);// 新增节点。颜色红色给红色cur->_col = RED;if (parent->_kv.first < kv.first){parent->_right = cur;}else{parent->_left = cur;}cur->_parent = parent;while (parent && parent->_col == RED) //当父亲节点为红色，则出现了连续的红色，不符合条件{Node* grandfather = parent->_parent;//    g//  p   uif (parent == grandfather->_left) {Node* uncle = grandfather->_right;if (uncle && uncle->_col == RED) //叔叔存在并且为红{parent->_col = uncle->_col = BLACK;grandfather->_col = RED;cur = grandfather;parent = cur->_parent; //往上面走}else{//u存在且为黑或不存在 ->变色再继续往上处理 + 变色if (cur == parent->_left) { //cur存在那么cur一定为红色 //    g//  p   u//c//单旋，把p旋转上去，p作为子树根节点，g作为p的右RotateR(grandfather);parent->_col = BLACK;grandfather->_col = RED;}else{//    g//  p   u//    c//双旋，将cur旋转上去，p作为cur的左，然后再旋转把cur旋转上去，g作为cur右边RotateL(parent);RotateR(grandfather);cur->_col = BLACK;grandfather->_col = RED;}break;}}else{//    g//  u   pNode* 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; //无论什么情况根节点都为黑return true;}void InOrder(){_InOrder(_root);cout << endl;}int Height(){return _Height(_root);}int Size(){return _Size(_root);}Node* Find(const K& key){Node* cur = _root;while (cur){if (cur->_kv.first < key){cur = cur->_right;}else if (cur->_kv.first > key){cur = cur->_left;}else{return cur;}}return nullptr;}bool IsBalance(){if (_root == nullptr)return true;if (_root->_col == RED){return false;}// 随便找条路径作为参考值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);}int _Size(Node* root){return root == nullptr ? 0 : _Size(root->_left) + _Size(root->_right) + 1;}int _Height(Node* root){if (root == nullptr)return 0;int leftHeight = _Height(root->_left);int rightHeight = _Height(root->_right);return leftHeight > rightHeight ? leftHeight + 1 : rightHeight + 1;}void _InOrder(Node* root){if (root == nullptr){return;}_InOrder(root->_left);cout << root->_kv.first << ":" << root->_kv.second << endl;_InOrder(root->_right);}void RotateL(Node* parent){_rotateNum++;Node* subR = parent->_right;Node* subRL = subR->_left;parent->_right = subRL;if (subRL)subRL->_parent = parent;Node* parentParent = parent->_parent;subR->_left = parent;parent->_parent = subR;if (parentParent == nullptr){_root = subR;subR->_parent = nullptr;}else{if (parent == parentParent->_left){parentParent->_left = subR;}else{parentParent->_right = subR;}subR->_parent = parentParent;}}void  RotateR(Node* parent){_rotateNum++;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;subL->_parent = nullptr;}else{if (parent == parentParent->_left){parentParent->_left = subL;}else{parentParent->_right = subL;}subL->_parent = parentParent;}}void Destroy(Node* root){if (root == nullptr)return;Destroy(root->_left);Destroy(root->_right);delete root;}Node* Copy(Node* root){if (root == nullptr) return nullptr;Node* newRoot = new Node(root->_kv);newRoot->_left = Copy(root->_left);newRoot->_right = Copy(root->_right);return newRoot;}private:Node* _root = nullptr;public:int _rotateNum = 0;
};void TestRBTree1()
{RBTree<int, int> t;//int a[] = { 16, 3, 7, 11, 9, 26, 18, 14, 15 };int a[] = { 4, 2, 6, 1, 3, 5, 15, 7, 16, 14 };for (auto e : a){t.Insert({ e, e });}t.InOrder();cout << t.IsBalance() << endl;
}