C++ 红黑树实现详解：规则、结构与插入查找验证

设 c 为当前节点，p 为父节点，g 为祖父节点，u 为叔叔节点。

• 情况 1：变色 若 u 存在且为红色，则将 p 和 u 变黑，g 变红。把 g 当做新的 c，继续往上更新。这是因为 p 和 u 变黑增加了左/右子树的黑高，g 变红抵消了变化，保持整棵树黑高不变，同时解决了 c 和 p 连续红的问题。

• 情况 2：单旋 + 变色 若 u 不存在或为黑色，单纯的变色无法解决问题，需要旋转。例如 p 是 g 的左孩子，c 是 p 的左孩子，以 g 为旋转点进行右单旋，再把 p 变黑，g 变红。

• 情况 3：双旋 + 变色 若 p 是 g 的左孩子，c 是 p 的右孩子，先以 p 为旋转点左单旋，再以 g 为旋转点右单旋，最后变色处理。

2.3 红黑树的插入代码实现

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; // 已存在
        }
    }

    cur = new Node(kv);
    cur->_col = RED;
    if (parent->_kv.first < kv.first) {
        parent->_right = cur;
    } else {
        parent->_left = cur;
    }
    cur->_parent = parent;

    while (parent && parent->_col == RED) {
        Node* grandfather = parent->_parent;
        if (grandfather->_left == parent) {
            Node* uncle = grandfather->_right;
            // 情况一：叔叔存在且为红
            if (uncle && uncle->_col == RED) {
                parent->_col = uncle->_col = BLACK;
                grandfather->_col = RED;
                cur = grandfather;
                parent = cur->_parent;
            } else {
                // 情况二：叔叔不存在或为黑
                if (cur == parent->_left) {
                    RotateR(grandfather);
                    parent->_col = BLACK;
                    grandfather->_col = RED;
                } else {
                    RotateL(parent);
                    RotateR(grandfather);
                    cur->_col = BLACK;
                    grandfather->_col = RED;
                }
                break;
            }
        } else {
            Node* uncle = grandfather->_left;
            // 情况一：叔叔存在且为红
            if (uncle && uncle->_col == RED) {
                parent->_col = uncle->_col = BLACK;
                grandfather->_col = RED;
                cur = grandfather;
                parent = cur->_parent;
            } else {
                // 情况二：叔叔不存在或为黑
                if (cur == parent->_right) {
                    RotateL(grandfather);
                    parent->_col = BLACK;
                    grandfather->_col = RED;
                } else {
                    RotateR(parent);
                    RotateL(grandfather);
                    cur->_col = BLACK;
                    grandfather->_col = RED;
                }
                break;
            }
        }
    }
    _root->_col = BLACK;
    return true;
}

2.4 红黑树的查找

按二叉搜索树逻辑实现即可，搜索效率为 O(logN)。

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;
}

2.5 红黑树的验证

检查最长路径不超过最短路径的两倍是不可行的，因为即使满足条件也可能颜色不合规。我们直接检查 4 点规则。

1. 规则 1 天然实现。
2. 规则 2 直接检查根节点。
3. 规则 3 前序遍历，遇到红色结点检查父亲颜色。
4. 规则 4 前序遍历，记录路径上的黑色结点数量，比较是否一致。

bool Check(Node* root, int blackNum, const int refNum) {
    if (root == nullptr) {
        if (refNum != blackNum) {
            cout << "存在黑色结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 << "存在连续的红色结点" << endl;
        return false;
    }
    if (root->_col == BLACK) {
        blackNum++;
    }
    return Check(root->_left, blackNum, refNum) && Check(root->_right, blackNum, refNum);
}

bool IsBalance() {
    if (_root == nullptr) return true;
    if (_root->_col == RED) return false;
    int refNum = 0;
    Node* cur = _root;
    while (cur) {
        if (cur->_col == BLACK) ++refNum;
        cur = cur->_left;
    }
    return Check(_root, 0, refNum);
}

2.6 红黑树的删除

红黑树的删除较为复杂，本章节不做详细讲解，有兴趣的同学可参考《算法导论》或《STL 源码剖析》。

三、完整代码

以下是完整的头文件及测试代码，包含红黑树与 AVL 树的对比实现。

RBTree.h

#pragma once
#include <iostream>
#include <utility>
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:
    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);
        cur->_col = RED;
        if (parent->_kv.first < kv.first) {
            parent->_right = cur;
        } else {
            parent->_left = cur;
        }
        cur->_parent = parent;

        while (parent && parent->_col == RED) {
            Node* grandfather = parent->_parent;
            if (grandfather->_left == parent) {
                Node* uncle = grandfather->_right;
                if (uncle && uncle->_col == RED) {
                    parent->_col = uncle->_col = BLACK;
                    grandfather->_col = RED;
                    cur = grandfather;
                    parent = cur->_parent;
                } else {
                    if (cur == parent->_left) {
                        RotateR(grandfather);
                        parent->_col = BLACK;
                        grandfather->_col = RED;
                    } else {
                        RotateL(parent);
                        RotateR(grandfather);
                        cur->_col = BLACK;
                        grandfather->_col = RED;
                    }
                    break;
                }
            } else {
                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->_right) {
                        RotateL(grandfather);
                        parent->_col = BLACK;
                        grandfather->_col = RED;
                    } else {
                        RotateR(parent);
                        RotateL(grandfather);
                        cur->_col = BLACK;
                        grandfather->_col = RED;
                    }
                    break;
                }
            }
        }
        _root->_col = BLACK;
        return true;
    }

    void InOrder() {
        _InOrder(_root);
        cout << endl;
    }

    bool Check(Node* root, int blackNum, const int refNum) {
        if (root == nullptr) {
            if (refNum != blackNum) {
                cout << "存在黑色结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 << "存在连续的红色结点" << endl;
            return false;
        }
        if (root->_col == BLACK) {
            blackNum++;
        }
        return Check(root->_left, blackNum, refNum) && Check(root->_right, blackNum, refNum);
    }

    bool IsBalance() {
        if (_root == nullptr) return true;
        if (_root->_col == RED) return false;
        int refNum = 0;
        Node* cur = _root;
        while (cur) {
            if (cur->_col == BLACK) ++refNum;
            cur = cur->_left;
        }
        return Check(_root, 0, refNum);
    }

    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;
    }

protected:
    int _Size(Node* root) {
        if (root == nullptr) return 0;
        return _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 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 (parent == _root) {
            _root = subL;
            subL->_parent = nullptr;
        } else {
            if (ppNode->_left == parent) {
                ppNode->_left = subL;
            } else {
                ppNode->_right = subL;
            }
            subL->_parent = ppNode;
        }
    }
    void RotateL(Node* parent) {
        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 _InOrder(Node* root) {
        if (root == nullptr) return;
        _InOrder(root->_left);
        cout << root->_kv.first << ":" << root->_kv.second << endl;
        _InOrder(root->_right);
    }

private:
    Node* _root = nullptr;
};

AVLTree.h

#pragma once
#include <iostream>
#include <cmath>
#include <cstdlib>
#include <cassert>
using namespace std;

template<class K, class V>
struct AVLTreeNode {
    pair<K, V> _kv;
    AVLTreeNode<K, V>* _left;
    AVLTreeNode<K, V>* _right;
    AVLTreeNode<K, V>* _parent;
    int _bf;

    AVLTreeNode(const pair<K, V>& kv)
        :_kv(kv), _left(nullptr), _right(nullptr), _parent(nullptr), _bf(0) {}
};

template<class K, class V>
class AVLTree {
    typedef AVLTreeNode<K, V> Node;
public:
    bool Insert(const pair<K, V>& kv) {
        if (_root == nullptr) {
            _root = new Node(kv);
            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;
        } else {
            parent->_left = cur;
        }
        cur->_parent = parent;

        while (parent) {
            if (cur == parent->_right) {
                parent->_bf++;
            } else {
                parent->_bf--;
            }
            if (parent->_bf == 0) {
                break;
            } else if (parent->_bf == 1 || parent->_bf == -1) {
                cur = parent;
                parent = parent->_parent;
            } else if (parent->_bf == 2 || parent->_bf == -2) {
                if (parent->_bf == -2 && cur->_bf == -1) {
                    RotateR(parent);
                } else if (parent->_bf == 2 && cur->_bf == 1) {
                    RotateL(parent);
                } else if (parent->_bf == -2 && cur->_bf == 1) {
                    RotateLR(parent);
                } else if (parent->_bf == 2 && cur->_bf == -1) {
                    RotateRL(parent);
                } else {
                    assert(false);
                }
                break;
            } else {
                assert(false);
            }
        }
        return true;
    }

    void InOrder() {
        _InOrder(_root);
        cout << endl;
    }

    bool IsBalanceTree() {
        return _IsBalanceTree(_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;
    }

    int Height() { return _Height(_root); }
    int Size() { return _Size(_root); }

private:
    int _Size(Node* root) {
        if (root == nullptr) return 0;
        return _Size(root->_left) + _Size(root->_right) + 1;
    }
    bool _IsBalanceTree(Node* root) {
        if (nullptr == root) return true;
        int leftHeight = _Height(root->_left);
        int rightHeight = _Height(root->_right);
        int diff = rightHeight - leftHeight;
        if (abs(diff) >= 2) {
            cout << root->_kv.first << "高度差异常" << endl;
            return false;
        }
        if (root->_bf != diff) {
            cout << root->_kv.first << "平衡因子异常" << endl;
            return false;
        }
        return _IsBalanceTree(root->_left) && _IsBalanceTree(root->_right);
    }
    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 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 (parent == _root) {
            _root = subL;
            subL->_parent = nullptr;
        } else {
            if (ppNode->_left == parent) {
                ppNode->_left = subL;
            } else {
                ppNode->_right = subL;
            }
            subL->_parent = ppNode;
        }
        parent->_bf = 0;
        subL->_bf = 0;
    }
    void RotateL(Node* parent) {
        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;
        }
        parent->_bf = subR->_bf = 0;
    }
    void RotateLR(Node* parent) {
        Node* subL = parent->_left;
        Node* subLR = subL->_right;
        int bf = subLR->_bf;
        RotateL(parent->_left);
        RotateR(parent);
        if (bf == -1) {
            subL->_bf = 0;
            parent->_bf = 1;
            subLR->_bf = 0;
        } else if (bf == 1) {
            subL->_bf = -1;
            parent->_bf = 0;
            subLR->_bf = 0;
        } else if (bf == 0) {
            subL->_bf = 0;
            parent->_bf = 0;
            subLR->_bf = 0;
        } else {
            assert(false);
        }
    }
    void RotateRL(Node* parent) {
        Node* subR = parent->_right;
        Node* subRL = subR->_left;
        int bf = subRL->_bf;
        RotateR(parent->_right);
        RotateL(parent);
        if (bf == 0) {
            subR->_bf = 0;
            subRL->_bf = 0;
            parent->_bf = 0;
        } else if (bf == 1) {
            subR->_bf = 0;
            subRL->_bf = 0;
            parent->_bf = -1;
        } else if (bf == -1) {
            subR->_bf = 1;
            subRL->_bf = 0;
            parent->_bf = 0;
        } else {
            assert(false);
        }
    }

private:
    Node* _root = nullptr;
};

test.cpp

#define _CRT_SECURE_NO_WARNINGS 1
#include<iostream>
#include<vector>
#include<assert.h>
#include<time.h>
#include<cstdlib>
using namespace std;
#include"RBTree.h"
#include"AVLTree.h"

void TestRBTree1() {
    RBTree<int, int> t;
    int a[] = { 16, 3, 7, 11, 9, 26, 18, 14, 15 };
    for (auto e : a) {
        t.Insert({ e, e });
        cout << "Insert:" << e << "->";
        cout << t.IsBalance() << endl;
    }
    t.InOrder();
    cout << t.IsBalance() << endl;
}

void TestRBTree2() {
    const int N = 10000000;
    vector<int> v;
    v.reserve(N);
    srand(time(0));
    for (size_t i = 0; i < N; i++) {
        v.push_back(rand() + i);
    }
    size_t begin2 = clock();
    RBTree<int, int> t;
    for (auto e : v) {
        t.Insert(make_pair(e, e));
    }
    size_t end2 = clock();
    cout << t.IsBalance() << endl;
    cout << "RBTree Insert:" << end2 - begin2 << endl;
    cout << "RBTree Height:" << t.Height() << endl;
    cout << "RBTree Size:" << t.Size() << endl;
    size_t begin1 = clock();
    for (size_t i = 0; i < N; i++) {
        t.Find((rand() + i));
    }
    size_t end1 = clock();
    cout << "RBTree Find:" << end1 - begin1 << endl << endl;
}

void TestAVLTree2() {
    const int N = 10000000;
    vector<int> v;
    v.reserve(N);
    srand(time(0));
    for (size_t i = 0; i < N; i++) {
        v.push_back(rand() + i);
    }
    size_t begin2 = clock();
    AVLTree<int, int> t;
    for (auto e : v) {
        t.Insert(make_pair(e, e));
    }
    size_t end2 = clock();
    cout << t.IsBalanceTree() << endl;
    cout << "AVLTree Insert:" << end2 - begin2 << endl;
    cout << "AVLTree Height:" << t.Height() << endl;
    cout << "AVLTree Size:" << t.Size() << endl;
    size_t begin1 = clock();
    for (size_t i = 0; i < N; i++) {
        t.Find((rand() + i));
    }
    size_t end1 = clock();
    cout << "AVLTree Find:" << end1 - begin1 << endl;
}

int main() {
    TestRBTree2();
    TestAVLTree2();
    return 0;
}