C++ 红黑树：原理、旋转与完整实现

3.3 旋转操作

旋转是维持树结构的关键，分为左旋和右旋。旋转过程中需注意更新父子指针关系，特别是根节点的更新。

// 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;
    }
}

3.4 辅助功能设计

为了封装性，Size()、Height() 等功能采用双重接口设计：公共接口不传参数，私有递归接口接收当前节点指针。

public:
    int Size() { return _Size(_root); }
private:
    int _Size(Node* root) {
        if (root == nullptr) return 0;
        return _Size(root->_left) + _Size(root->_right) + 1;
    }

3.5 验证与测试

验证红黑树需检查四条规则，特别是黑色节点数量的一致性。可通过前序遍历统计路径上的黑色节点数，并与最左侧路径进行比较。

4. 完整代码示例

以下是 RBTree.h 的核心实现片段，包含插入、旋转、查找及验证逻辑。

#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;

    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);
                        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;
    }

private:
    Node* _root = nullptr;
};

5. 运行演示

通过测试用例验证插入后的中序遍历结果及平衡性。对于大规模随机数据插入，可观察树的高度和执行耗时，验证其 O(logn) 的性能表现。

6. 总结

红黑树通过颜色约束和旋转操作实现了高效的自平衡。相比于 AVL 树，它在保持查询性能的同时减少了维护平衡的开销，是实际开发中首选的平衡树结构。建议读者参考《算法导论》或《STL 源码剖析》深入理解其删除操作的复杂逻辑。