C++ 红黑树深度解析：原理、旋转与完整实现

情况 3：双旋 + 变色

• 若 c 是 p 的右孩子，p 是 g 的左孩子（LR 型），先以 p 为支点左旋，再以 g 为支点右旋，c 变黑，g 变红。
• 若 c 是 p 的左孩子，p 是 g 的右孩子（RL 型），先以 p 为支点右旋，再以 g 为支点左旋，c 变黑，g 变红。

2.3 旋转操作

旋转是维持树结构的关键。注意处理 parent 指针和根节点的更新。

// 右旋示例
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 (parent == _root) {
        _root = subL;
        subL->_parent = nullptr;
    } else {
        if (parentParent->_left == parent)
            parentParent->_left = subL;
        else
            parentParent->_right = subL;
        subL->_parent = parentParent;
    }
}

2.4 接口设计

对于 Size()、Height() 等函数，通常采用双重接口设计：

• 公共接口：用户调用 tree.Size()，不需要传参，安全易用。
• 私有递归接口：内部实现 _Size(root)，需要传入当前节点以支持递归遍历。

这种设计避免了暴露内部根节点，同时保证了封装性。

2.5 验证与测试

验证红黑树是否合法，需检查四条规则。特别是规则 4（路径黑高一致），可以通过前序遍历记录路径上的黑色节点数来比较。

完整代码示例

RBTree.h

#pragma once
#include<iostream>
using namespace std;
#include<assert.h>

enum Color { BLACK, RED };

template<class K, class V>
struct RBTreeNode {
    pair<K, V> _kv;
    RBTreeNode* _left;
    RBTreeNode* _right;
    RBTreeNode* _parent;
    Color _col;

    RBTreeNode(const pair<K, V>& kv)
        :_kv(kv), _left(nullptr), _right(nullptr), _parent(nullptr), _col(RED) {}
};

template<class K, class V>
struct 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; // Key exists
            }
        }

        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);
                        grandfather->_col = RED;
                    }
                    break;
                }
            }
        }
        _root->_col = BLACK;
        return true;
    }

    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 (parent == _root) {
            _root = subL;
            subL->_parent = nullptr;
        } else {
            if (parentParent->_left == parent)
                parentParent->_left = subL;
            else
                parentParent->_right = subL;
            subL->_parent = parentParent;
        }
    }

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

    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 IsBalanceTree() {
        if (_root && _root->_col == RED) return false;
        int left_bn = 0;
        Node* cur = _root;
        while (cur) {
            if (cur->_col == BLACK) left_bn++;
            cur = cur->_left;
        }
        return _Checkcolor(_root, 0, left_bn);
    }

    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;
    }

    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;
    }

    bool _Checkcolor(Node* root, int root_cur_bn, const int left_bn) {
        if (root == nullptr) {
            if (root_cur_bn != left_bn) {
                cout << "黑色节点的数量不相等" << endl;
                return false;
            }
            return true;
        }
        if (root->_col == BLACK) root_cur_bn++;
        if (root->_col == RED && root->_parent && root->_parent->_col == RED) {
            cout << root->_kv.first << "存在连续的红色节点" << endl;
            return false;
        }
        return _Checkcolor(root->_left, root_cur_bn, left_bn) &&
               _Checkcolor(root->_right, root_cur_bn, left_bn);
    }

    int _InOrder(Node* root) {
        if (root == nullptr) return 0;
        _InOrder(root->_left);
        cout << root->_kv.first << " ";
        _InOrder(root->_right);
        return 0;
    }

    Node* _root = nullptr;
};

Tree.cpp

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

void TestRBTree1() {
    RBTree<int, int> t;
    int a[] = { 4,2,6,1,3,5,15,7,16,14 };
    for (auto e : a) {
        t.Insert({ e,e });
        cout << "Insert:" << e << "->";
        t.InOrder();
    }
    t.InOrder();
}

void TestRBTree2() {
    const int N = 1000000;
    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 << "Insert:" << end2 - begin2 << endl;
    cout << t.IsBalanceTree() << endl;
    cout << "Height:" << t.Height() << endl;
    cout << "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 << "Find:" << end1 - begin1 << endl;
}

int main() {
    //TestRBTree1();
    TestRBTree2();
    return 0;
}

运行上述代码可以验证红黑树的插入性能及平衡性。在百万级数据插入场景下，红黑树依然能保持稳定的对数级高度，验证了其工程价值。