深入理解红黑树：在C++中实现插入、删除和查找操作

红黑树是一种自平衡二叉搜索树，广泛应用于各种算法和系统中。它通过颜色属性和旋转操作来保持树的平衡，从而保证插入、删除和查找操作的时间复杂度为O(log n)。本文将详细介绍如何在C++中实现一个红黑树，并提供插入、删除和查找操作的具体实现。

红黑树的基本性质

红黑树具有以下性质：

每个节点要么是红色，要么是黑色。
根节点是黑色。
每个叶子节点（NIL节点）是黑色。
如果一个节点是红色的，则它的两个子节点都是黑色的（即没有两个连续的红色节点）。
对每个节点，从该节点到其所有后代叶子节点的路径上，包含相同数量的黑色节点。

这些性质确保了红黑树的平衡性，使得树的最长路径不会超过最短路径的两倍。

红黑树节点定义

首先，我们定义一个红黑树节点类，用于表示红黑树中的每个节点。

cpp 复制代码

enum Color { RED, BLACK };

template <typename T>
class Node {
public:
    T data;
    Color color;
    Node* left;
    Node* right;
    Node* parent;

    Node(T data) : data(data), color(RED), left(nullptr), right(nullptr), parent(nullptr) {}
};

红黑树类定义

接下来，我们定义一个红黑树类，包含红黑树的基本结构和成员函数。

cpp 复制代码

template <typename T>
class RedBlackTree {
private:
    Node<T>* root;

    void rotateLeft(Node<T>*& root, Node<T>*& pt);
    void rotateRight(Node<T>*& root, Node<T>*& pt);
    void fixInsert(Node<T>*& root, Node<T>*& pt);
    void fixDelete(Node<T>*& root, Node<T>*& pt);
    void inorderHelper(Node<T>* root);
    Node<T>* BSTInsert(Node<T>* root, Node<T>* pt);
    Node<T>* minValueNode(Node<T>* node);
    Node<T>* deleteBST(Node<T>* root, T data);

public:
    RedBlackTree() : root(nullptr) {}

    void insert(const T& data);
    void deleteNode(const T& data);
    bool search(const T& data);
    void inorder();
};

插入操作

插入操作包括标准的二叉搜索树插入和红黑树的修复操作。首先，我们进行标准的BST插入，然后通过旋转和重新着色来修复红黑树的性质。

cpp 复制代码

template <typename T>
void RedBlackTree<T>::insert(const T& data) {
    Node<T>* pt = new Node<T>(data);
    root = BSTInsert(root, pt);
    fixInsert(root, pt);
}

template <typename T>
Node<T>* RedBlackTree<T>::BSTInsert(Node<T>* root, Node<T>* pt) {
    if (root == nullptr) return pt;

    if (pt->data < root->data) {
        root->left = BSTInsert(root->left, pt);
        root->left->parent = root;
    } else if (pt->data > root->data) {
        root->right = BSTInsert(root->right, pt);
        root->right->parent = root;
    }

    return root;
}

template <typename T>
void RedBlackTree<T>::fixInsert(Node<T>*& root, Node<T>*& pt) {
    Node<T>* parent_pt = nullptr;
    Node<T>* grand_parent_pt = nullptr;

    while ((pt != root) && (pt->color != BLACK) && (pt->parent->color == RED)) {
        parent_pt = pt->parent;
        grand_parent_pt = pt->parent->parent;

        if (parent_pt == grand_parent_pt->left) {
            Node<T>* uncle_pt = grand_parent_pt->right;

            if (uncle_pt != nullptr && uncle_pt->color == RED) {
                grand_parent_pt->color = RED;
                parent_pt->color = BLACK;
                uncle_pt->color = BLACK;
                pt = grand_parent_pt;
            } else {
                if (pt == parent_pt->right) {
                    rotateLeft(root, parent_pt);
                    pt = parent_pt;
                    parent_pt = pt->parent;
                }

                rotateRight(root, grand_parent_pt);
                std::swap(parent_pt->color, grand_parent_pt->color);
                pt = parent_pt;
            }
        } else {
            Node<T>* uncle_pt = grand_parent_pt->left;

            if (uncle_pt != nullptr && uncle_pt->color == RED) {
                grand_parent_pt->color = RED;
                parent_pt->color = BLACK;
                uncle_pt->color = BLACK;
                pt = grand_parent_pt;
            } else {
                if (pt == parent_pt->left) {
                    rotateRight(root, parent_pt);
                    pt = parent_pt;
                    parent_pt = pt->parent;
                }

                rotateLeft(root, grand_parent_pt);
                std::swap(parent_pt->color, grand_parent_pt->color);
                pt = parent_pt;
            }
        }
    }

    root->color = BLACK;
}

删除操作

删除操作相对复杂，需要考虑多种情况。首先，我们进行标准的BST删除，然后通过旋转和重新着色来修复红黑树的性质。

cpp 复制代码

template <typename T>
void RedBlackTree<T>::deleteNode(const T& data) {
    Node<T>* node = deleteBST(root, data);
    if (node != nullptr) {
        fixDelete(root, node);
    }
}

template <typename T>
Node<T>* RedBlackTree<T>::deleteBST(Node<T>* root, T data) {
    if (root == nullptr) return root;

    if (data < root->data) {
        return deleteBST(root->left, data);
    } else if (data > root->data) {
        return deleteBST(root->right, data);
    }

    if (root->left == nullptr || root->right == nullptr) {
        return root;
    }

    Node<T>* temp = minValueNode(root->right);
    root->data = temp->data;
    return deleteBST(root->right, temp->data);
}

template <typename T>
void RedBlackTree<T>::fixDelete(Node<T>*& root, Node<T>*& pt) {
    Node<T>* sibling;

    while (pt != root && pt->color == BLACK) {
        if (pt == pt->parent->left) {
            sibling = pt->parent->right;

            if (sibling->color == RED) {
                sibling->color = BLACK;
                pt->parent->color = RED;
                rotateLeft(root, pt->parent);
                sibling = pt->parent->right;
            }

            if (sibling->left->color == BLACK && sibling->right->color == BLACK) {
                sibling->color = RED;
                pt = pt->parent;
            } else {
                if (sibling->right->color == BLACK) {
                    sibling->left->color = BLACK;
                    sibling->color = RED;
                    rotateRight(root, sibling);
                    sibling = pt->parent->right;
                }

                sibling->color = pt->parent->color;
                pt->parent->color = BLACK;
                sibling->right->color = BLACK;
                rotateLeft(root, pt->parent);
                pt = root;
            }
        } else {
            sibling = pt->parent->left;

            if (sibling->color == RED) {
                sibling->color = BLACK;
                pt->parent->color = RED;
                rotateRight(root, pt->parent);
                sibling = pt->parent->left;
            }

            if (sibling->left->color == BLACK && sibling->right->color == BLACK) {
                sibling->color = RED;
                pt = pt->parent;
            } else {
                if (sibling->left->color == BLACK) {
                    sibling->right->color = BLACK;
                    sibling->color = RED;
                    rotateLeft(root, sibling);
                    sibling = pt->parent->left;
                }

                sibling->color = pt->parent->color;
                pt->parent->color = BLACK;
                sibling->left->color = BLACK;
                rotateRight(root, pt->parent);
                pt = root;
            }
        }
    }

    pt->color = BLACK;
}

查找操作

查找操作相对简单，通过比较目标值与当前节点的值，决定向左子树还是右子树移动，直到找到目标值或到达空节点。

cpp 复制代码

template <typename T>
bool RedBlackTree<T>::search(const T& data) {
    Node<T>* current = root;
    while (current != nullptr) {
        if (data == current->data) {
            return true;
        } else if (data < current->data) {
            current = current->left;
        } else {
            current = current->right;
        }
    }
    return false;
}

中序遍历

中序遍历用于验证红黑树的结构，确保所有节点按顺序排列。

cpp 复制代码

template <typename T>
void RedBlackTree<T>::inorder() {
    inorderHelper(root);
}

template <typename T>
void RedBlackTree<T>::inorderHelper(Node<T>* root) {
    if (root == nullptr) return;

    inorderHelper(root->left);