Cây AVL: Xoay, chèn, xóa bằng C++ Ví dụ

Bởi: Sarah Chen Sarah Chen
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⚡ Tóm tắt thông minh

AVL Trees are self-balancing binary search trees where the height difference between the left and right subtrees of every node stays within -1, 0, or +1, guaranteeing O(log n) search performance.

  • 🌲 Định nghĩa: A binary search tree in which the balance factor of every node lies in {-1, 0, +1}, named after inventors Adelson-Velsky and Landis.
  • ⚖️ Balance Factor: Computed as height(left) − height(right); values outside {-1, 0, +1} trigger a rotation to restore balance.
  • 🔄 Vòng quay: Four cases — LL, RR, LR, and RL — realign nodes after unbalanced insertions or deletions to keep the tree logarithmic in height.
  • ➕ Chèn: Standard BST insert followed by an upward walk that recomputes balance factors and performs at most one single or double rotation.
  • ➖ Xóa: Same as BST deletion but may cascade multiple rotations up the tree because subtree height can shrink at every ancestor.
  • 🚀 Ứng dụng Databases, in-memory indexes, filesystem metadata, and AI search structures use AVL Trees for fast ordered lookups.
Đọc thêm

Cây AVL

Cây AVL là gì?

Cây AVL are binary search trees in which the height difference between the left and right subtree of every node is -1, 0, or +1. They are self-balancing BSTs that maintain logarithmic search time, named after inventors Adelson-Velsky and Landis (AVL).

Cây AVL hoạt động như thế nào?

To understand why AVL Trees exist, look at what goes wrong with a plain Cây tìm kiếm nhị phân. Consider these keys inserted in the given order:

Cây AVL hoạt động

Trực quan hóa cây AVL

The tree grows linearly when keys arrive in increasing order, degenerating search to O(n). That defeats the purpose of a BST — only a balanced tree keeps search logarithmic. Now look at the same keys inserted in a different order.

Cây AVL hoạt động

Same keys, different insertion order produces a shallower shape, so every search runs in O(log n). AVL Trees enforce that shape by watching the height on every insertion and correcting imbalance without breaking BST ordering.

Hệ số cân bằng trong cây AVL

The balance factor (BF) tracks each node’s height so the tree can self-balance on the fly.

Thuộc tính của yếu tố cân bằng

Hệ số cân bằng trong cây AVL

Cây AVL hệ số cân bằng

  • The balance factor is the difference between the height of the left subtree and the height of the right subtree.
  • Balance factor(node) = height(node->left) − height(node->right)
  • The only allowed values are −1, 0, and +1.
  • A value of −1 means the right subtree contains one extra level — the node is right-heavy.
  • A value of +1 means the left subtree contains one extra level — the node is left-heavy.
  • A value of 0 means both sides have equal height — the node is perfectly balanced.

Xoay AVL

Rotations run whenever an insertion or deletion breaks the balance factor rule. The four cases are LL, RR, LR, and RL.

Xoay trái – Xoay trái

Việc xoay này được thực hiện khi một nút mới được chèn vào nút con bên trái của cây con bên trái.

Cây AVL Trái – Xoay Trái

Cây AVL Trái – Xoay Trái

A single right rotation is performed. This case fires when a node has BF +2 and its left child has BF +1.

Phải – Xoay Phải

Việc xoay này được thực hiện khi một nút mới được chèn vào con bên phải của cây con bên phải.

Cây AVL phải – Xoay phải

A single left rotation is performed. This case fires when a node has BF −2 and its right child has BF −1.

Xoay phải – trái

Việc xoay này được thực hiện khi một nút mới được chèn vào con trái của cây con bên phải.

Cây AVL Phải – Xoay Trái

Fires when BF(node) = −2 and BF(right-child) = +1. Right-rotate the right child, then left-rotate the node.

Xoay Trái – Phải

Việc xoay này được thực hiện khi một nút mới được chèn vào nút con bên phải của cây con bên trái.

Cây AVL Xoay Trái – Phải

Fires when BF(node) = +2 and BF(left-child) = −1. Left-rotate the left child, then right-rotate the node.

Chèn vào cây AVL

Insertion is almost identical to a plain BST insert. After every insert, the tree walks up and re-balances. Insert runs in O(log n) worst-case time.

Chèn vào cây AVL

Triển khai chèn cây AVL

Bước 1: Insert the node using the standard BST algorithm. In the example above, insert 160.

Bước 2: Update the balance factor of every ancestor along the insertion path.

Bước 3: If any ancestor violates the balance factor range, perform the matching rotation. In the example, node 350’s balance factor is violated, so an LL rotation restores balance.

  1. If BF(node) = +2 và BF(left-child) = +1, perform LL rotation.
  2. If BF(node) = −2 và BF(right-child) = −1, perform RR rotation.
  3. If BF(node) = −2 và BF(right-child) = +1, perform RL rotation.
  4. If BF(node) = +2 và BF(left-child) = −1, perform LR rotation.

NHỮNG BÀI VIẾT LIÊN QUAN

Xóa trong cây AVL

Deletion follows the same logic as a plain BST and re-balances afterwards.

Bước 1: Tìm phần tử trong cây.

Bước 2: Delete the node using standard BST deletion.

Bước 3: Two cases are possible.

Trường hợp 1: Xóa từ cây con bên phải.

  • 1A. If BF(node) = +2 và BF(left-child) = +1, perform LL rotation.
  • Năm 1B. If BF(node) = +2 và BF(left-child) = −1, perform LR rotation.
  • 1C. If BF(node) = +2 và BF(left-child) = 0, perform LL rotation.

Xóa trong cây AVL

Trường hợp 2: Deleting from the left subtree.

  • 2A. If BF(node) = −2 và BF(right-child) = −1, perform RR rotation.
  • Năm 2B. If BF(node) = −2 và BF(right-child) = +1, perform RL rotation.
  • 2C. If BF(node) = −2 và BF(right-child) = 0, perform RR rotation.

Xóa trong cây AVL

C++ Ví dụ về cây AVL

Dưới đây là một C++ program implementing AVL Trees:

#include <iostream>
#include <queue>
#include <unordered_map>
using namespace std;

struct node {
    struct node *left;
    int data;
    int height;
    struct node *right;
};

class AVL {
public:
    struct node *root;

    AVL() {
        this->root = NULL;
    }

    int calheight(struct node *p) {
        if (p->left && p->right) {
            if (p->left->height < p->right->height)
                return p->right->height + 1;
            else
                return p->left->height + 1;
        }
        else if (p->left && p->right == NULL) {
            return p->left->height + 1;
        }
        else if (p->left == NULL && p->right) {
            return p->right->height + 1;
        }
        return 0;
    }

    int bf(struct node *n) {
        if (n->left && n->right)
            return n->left->height - n->right->height;
        else if (n->left && n->right == NULL)
            return n->left->height;
        else if (n->left == NULL && n->right)
            return -n->right->height;
        return 0;
    }

    struct node *llrotation(struct node *n) {
        struct node *p = n;
        struct node *tp = p->left;
        p->left = tp->right;
        tp->right = p;
        return tp;
    }

    struct node *rrrotation(struct node *n) {
        struct node *p = n;
        struct node *tp = p->right;
        p->right = tp->left;
        tp->left = p;
        return tp;
    }

    struct node *rlrotation(struct node *n) {
        struct node *p = n;
        struct node *tp = p->right;
        struct node *tp2 = p->right->left;
        p->right = tp2->left;
        tp->left = tp2->right;
        tp2->left = p;
        tp2->right = tp;
        return tp2;
    }

    struct node *lrrotation(struct node *n) {
        struct node *p = n;
        struct node *tp = p->left;
        struct node *tp2 = p->left->right;
        p->left = tp2->right;
        tp->right = tp2->left;
        tp2->right = p;
        tp2->left = tp;
        return tp2;
    }

    struct node *insert(struct node *r, int data) {
        if (r == NULL) {
            r = new struct node;
            r->data = data;
            r->left = r->right = NULL;
            r->height = 1;
            return r;
        }
        if (data < r->data)
            r->left = insert(r->left, data);
        else
            r->right = insert(r->right, data);

        r->height = calheight(r);

        if (bf(r) == 2 && bf(r->left) == 1)       r = llrotation(r);
        else if (bf(r) == -2 && bf(r->right) == -1) r = rrrotation(r);
        else if (bf(r) == -2 && bf(r->right) == 1)  r = rlrotation(r);
        else if (bf(r) == 2 && bf(r->left) == -1)   r = lrrotation(r);

        return r;
    }

    void levelorder_newline() {
        if (this->root == NULL) {
            cout << "\nEmpty tree\n";
            return;
        }
        levelorder_newline(this->root);
    }

    void levelorder_newline(struct node *v) {
        queue<struct node *> q;
        struct node *cur;
        q.push(v);
        q.push(NULL);
        while (!q.empty()) {
            cur = q.front();
            q.pop();
            if (cur == NULL && q.size() != 0) {
                cout << "\n";
                q.push(NULL);
                continue;
            }
            if (cur != NULL) {
                cout << " " << cur->data;
                if (cur->left != NULL)  q.push(cur->left);
                if (cur->right != NULL) q.push(cur->right);
            }
        }
    }

    struct node *deleteNode(struct node *p, int data) {
        if (p->left == NULL && p->right == NULL) {
            if (p == this->root) this->root = NULL;
            delete p;
            return NULL;
        }
        struct node *q;
        if (p->data < data)      p->right = deleteNode(p->right, data);
        else if (p->data > data) p->left  = deleteNode(p->left, data);
        else {
            if (p->left != NULL) {
                q = inpre(p->left);
                p->data = q->data;
                p->left = deleteNode(p->left, q->data);
            } else {
                q = insuc(p->right);
                p->data = q->data;
                p->right = deleteNode(p->right, q->data);
            }
        }

        if (bf(p) == 2 && bf(p->left) == 1)         p = llrotation(p);
        else if (bf(p) == 2 && bf(p->left) == -1)    p = lrrotation(p);
        else if (bf(p) == 2 && bf(p->left) == 0)     p = llrotation(p);
        else if (bf(p) == -2 && bf(p->right) == -1)  p = rrrotation(p);
        else if (bf(p) == -2 && bf(p->right) == 1)   p = rlrotation(p);
        else if (bf(p) == -2 && bf(p->right) == 0)   p = rrrotation(p);

        return p;
    }

    struct node *inpre(struct node *p) {
        while (p->right != NULL) p = p->right;
        return p;
    }

    struct node *insuc(struct node *p) {
        while (p->left != NULL) p = p->left;
        return p;
    }

    ~AVL() {}
};

int main() {
    AVL b;
    int c, x;
    do {
        cout << "\n1.Display levelorder on newline";
        cout << "\n2.Insert";
        cout << "\n3.Delete\n";
        cout << "\n0.Exit\n";
        cout << "\nChoice: ";
        cin >> c;
        switch (c) {
        case 1: b.levelorder_newline(); break;
        case 2:
            cout << "\nEnter no. "; cin >> x;
            b.root = b.insert(b.root, x);
            break;
        case 3:
            cout << "\nWhat to delete? "; cin >> x;
            b.root = b.deleteNode(b.root, x);
            break;
        case 0: break;
        }
    } while (c != 0);
}

Running example of the code above:

  1. Copy the code above and save it in a file named avl.cpp.
  2. Biên dịch mã:
g++ avl.cpp -o run
  1. Chạy mã.
./run

C++ Ví dụ về cây AVL

Ưu điểm của cây AVL

  • The height of the AVL Tree is always balanced and never grows beyond log N.
  • Search is faster than a plain Binary Search Tree because the tree cannot degenerate.
  • Self-balancing is automatic — no rebuild step is required.
  • Deterministic performance suits real-time systems and in-memory indexes.

Câu Hỏi Thường Gặp

An AVL Tree is a self-balancing binary search tree where the balance factor of every node stays in {-1, 0, +1}. Rotations restore this invariant on every insert or delete, keeping search, insert, and delete at O(log n).

The balance factor of a node equals height(left subtree) minus height(right subtree). Values must lie in {-1, 0, +1}. A balance factor of +2 or -2 signals that an insertion or deletion has unbalanced that node and a rotation is required.

The four rotations are LL, RR, LR, and RL. LL uses a single right rotation, RR uses a single left rotation, and LR and RL are double rotations that combine one rotation on the child with an opposite rotation on the node.

Insertion follows the standard BST rule, then the tree walks back up updating heights. If any ancestor breaks the balance rule, one single or double rotation restores balance. At most one rotation per insert is ever needed.

AVL Trees are strictly balanced with a balance factor of at most one, giving faster lookups. Red-Black Trees allow looser balance, which makes insert and delete cheaper but search slightly slower. Databases prefer red-black for write-heavy loads.

AVL Trees power in-memory database indexes, filesystem metadata, priority queues, phonebook lookups, spell checkers, and any workload that needs deterministic O(log n) search plus in-order traversal for range queries.

Yes. AI systems use AVL Trees for symbol tables, ordered feature stores, k-d tree balancing, and nearest-neighbour lookups on structured data. They also underpin ranked retrieval indexes in intelligent search pipelines.

Yes. GitHub Copilot and similar AI assistants scaffold insert, delete, and rotation routines in C++, Java, hoặc là Python, and generate unit tests that verify the balance factor invariant on every operation.

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