红黑树详解初版

---

一、红黑树核心性质

红黑树是自平衡二叉搜索树，需满足5条性质：

1. 节点非红即黑
2. 根节点必须为黑
3. 叶子节点（NIL）均为黑
4. 红色节点子节点必为黑（无连续红节点）
5. 任意节点到叶子路径的黑节点数相同（黑高度一致）

---

二、C++节点结构定义

enum Colour { RED, BLACK };template <typename K, typename V>
struct RBTreeNode {K key;V value;Colour color;RBTreeNode *left, *right, *parent;RBTreeNode(K k, V v) : key(k), value(v), color(RED), left(nullptr), right(nullptr), parent(nullptr) {}
};

节点默认红色以最小化插入调整代价

---

三、插入操作详解

步骤1：二叉搜索树插入

bool Insert(const pair<K, V>& kv) {if (!root) { root = new Node(kv, BLACK); return true; }// 查找插入位置（同普通BST）//...// 新节点插入为红色cur = new Node(kv);cur->color = RED;parent->left/right = cur;cur->parent = parent;// 进入调整逻辑FixInsert(cur);
}

步骤2：插入后调整（FixInsert）

情况1：叔节点为红

处理方案：
父、叔变黑，祖父变红，递归处理祖父节点

parent->color = uncle->color = BLACK;
grandfather->color = RED;
cur = grandfather; // 向上递归

情况2：叔节点为黑（直线型）

处理方案：
祖父右旋，父变黑，祖父变红

RotateR(grandfather);
parent->color = BLACK;
grandfather->color = RED;

情况3：叔节点为黑（折线型）

处理方案：
先左旋父节点转换为直线型，再按情况2处理

RotateL(parent);
RotateR(grandfather);

---

四、删除操作核心逻辑

步骤1：BST标准删除

void Delete(K key) {Node* target = Search(key);if (!target) return;// 处理单子节点/叶子节点删除//...// 若删除黑色节点需进入双黑修正if (target->color == BLACK) FixDoubleBlack(replacement);
}

步骤2：双黑修正（FixDoubleBlack）

情况1：兄弟节点为红

处理方案：
父节点左旋，兄弟变黑，父变红

RotateL(parent);
sibling->color = BLACK;
parent->color = RED;

情况2：兄弟节点为黑且远侄子为红

处理方案：
父节点左旋，远侄子变黑，父继承原父颜色

RotateL(parent);
sibling->right->color = BLACK;
parent->color = original_color;

---

五、旋转操作代码实现

左旋示例：

关键步骤解析​：

1. 原父节点P的右子节点Y上升为新父节点
2. 将Y的左子树α挂载到X的右子树
3. 原父节点P的右指针指向Y
4. Y的左指针指向X（形成父子关系反转）

void RotateL(Node* x) {Node* y = x->right;x->right = y->left;if (y->left) y->left->parent = x;y->parent = x->parent;if (!x->parent) root = y;else if (x == x->parent->left) x->parent->left = y;else x->parent->right = y;y->left = x;x->parent = y;
}

右旋对称实现

---

六、示例

---

七、完整代码

#include <iostream>
using namespace std;enum Colour { RED, BLACK };template<class K, class V>
struct RBTreeNode {pair<K, V> _kv;Colour _col;RBTreeNode<K, V>* _left;RBTreeNode<K, V>* _right;RBTreeNode<K, V>* _parent;RBTreeNode(const pair<K, V>& kv): _kv(kv),_col(RED),_left(nullptr),_right(nullptr),_parent(nullptr) {}
};template<class K, class V>
class RBTree {typedef RBTreeNode<K, V> Node;
public:RBTree() {NIL = new Node(make_pair(K(), V()));NIL->_col = BLACK;_root = NIL;}bool Insert(const pair<K, V>& kv) {if (_root == NIL) {_root = new Node(kv);_root->_col = BLACK;_root->_left = _root->_right = NIL;return true;}Node* parent = NIL;Node* cur = _root;while (cur != NIL) {parent = cur;if (cur->_kv.first > kv.first)cur = cur->_left;else if (cur->_kv.first < kv.first)cur = cur->_right;elsereturn false;}Node* newNode = new Node(kv);newNode->_left = newNode->_right = NIL;if (parent->_kv.first > kv.first)parent->_left = newNode;elseparent->_right = newNode;newNode->_parent = parent;FixInsert(newNode);return true;}private:Node* _root;Node* NIL; // 哨兵节点[2,4]// 左旋修正void RotateL(Node* x) {Node* y = x->_right;x->_right = y->_left;if (y->_left != NIL)y->_left->_parent = x;y->_parent = x->_parent;if (x->_parent == NIL)_root = y;else if (x == x->_parent->_left)x->_parent->_left = y;elsex->_parent->_right = y;y->_left = x;x->_parent = y;}// 右旋修正void RotateR(Node* x) {Node* y = x->_left;x->_left = y->_right;if (y->_right != NIL)y->_right->_parent = x;y->_parent = x->_parent;if (x->_parent == NIL)_root = y;else if (x == x->_parent->_right)x->_parent->_right = y;elsex->_parent->_left = y;y->_right = x;x->_parent = y;}void FixInsert(Node* z) {while (z->_parent->_col == RED) {Node* g = z->_parent->_parent;if (z->_parent == g->_left) {Node* u = g->_right;if (u->_col == RED) { // Case 1z->_parent->_col = BLACK;u->_col = BLACK;g->_col = RED;z = g;} else {if (z == z->_parent->_right) { // Case 2z = z->_parent;RotateL(z);}// Case 3z->_parent->_col = BLACK;g->_col = RED;RotateR(g);}} else { // 对称处理[4,6]Node* u = g->_left;if (u->_col == RED) {z->_parent->_col = BLACK;u->_col = BLACK;g->_col = RED;z = g;} else {if (z == z->_parent->_left) {z = z->_parent;RotateR(z);}z->_parent->_col = BLACK;g->_col = RED;RotateL(g);}}if (z == _root) break;}_root->_col = BLACK; // 强制根节点为黑[4,8]}
};// 测试用例
int main() {RBTree<int, int> tree;// 测试插入序列（触发所有修复情况）int arr[] = {10, 20, 30, 40, 50, 60, 70, 80, 90};for (auto num : arr) {tree.Insert(make_pair(num, num));}return 0;
}

---

八、应用场景

1. C++ STL：map/set底层实现
2. Java集合：TreeMap的平衡机制
3. Linux内核：进程调度CFS算法