AVL stabla: rotacije, umetanje, brisanje sa C++ Primjer

⚡ Pametni sažetak

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.

🌲 Definicija: 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.
🔄 Rotacije: Four cases — LL, RR, LR, and RL — realign nodes after unbalanced insertions or deletions to keep the tree logarithmic in height.
➕ Umetanje: Standard BST insert followed by an upward walk that recomputes balance factors and performs at most one single or double rotation.
➖ Brisanje: Same as BST deletion but may cascade multiple rotations up the tree because subtree height can shrink at every ancestor.
🚀 Primjena: Databases, in-memory indexes, filesystem metadata, and AI search structures use AVL Trees for fast ordered lookups.

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Što su AVL stabla?

AVL stabla 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).

Kako radi AVL stablo?

To understand why AVL Trees exist, look at what goes wrong with a plain Stablo binarnog pretraživanja. Consider these keys inserted in the given order:

Vizualizacija AVL stabla

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.

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.

Faktor ravnoteže u AVL stablima

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

Svojstva faktora ravnoteže

Faktor ravnoteže AVL stablo

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.

AVL rotacije

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

Lijevo – Lijeva rotacija

Ova rotacija se izvodi kada se novi čvor umetne u lijevo dijete lijevog podstabla.

AVL stablo lijevo – rotacija lijevo

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

Desno – Desna rotacija

Ova rotacija se izvodi kada se novi čvor umetne u desno dijete desnog podstabla.

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

Rotacija desno – lijevo

Ova rotacija se izvodi kada se novi čvor umetne u lijevo dijete desnog podstabla.

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

Rotacija lijevo – desno

Ova rotacija se izvodi kada se novi čvor umetne u desno dijete lijevog podstabla.

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

Umetanje u AVL stabla

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.

Implementacija umetanja AVL stabla

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

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

Korak 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.

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

Brisanje u AVL stablima

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

Korak 1: Pronađite element u stablu.

Korak 2: Delete the node using standard BST deletion.

Korak 3: Two cases are possible.

Slučaj 1: Brisanje iz desnog podstabla.

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

Slučaj 2: Deleting from the left subtree.

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

C++ Primjer AVL stabala

Ispod je a 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:

Copy the code above and save it in a file named avl.cpp.
Sastavite kod:

g++ avl.cpp -o run

Pokrenite kod.

./run

Prednosti AVL stabala

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.

Pitanja i odgovori

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, ili Python, and generate unit tests that verify the balance factor invariant on every operation.

AVL stabla: rotacije, umetanje, brisanje sa C++ Primjer

Što su AVL stabla?

Kako radi AVL stablo?

Faktor ravnoteže u AVL stablima

Svojstva faktora ravnoteže

AVL rotacije

Lijevo – Lijeva rotacija

Desno – Desna rotacija

Rotacija desno – lijevo

Rotacija lijevo – desno

Umetanje u AVL stabla

Brisanje u AVL stablima

C++ Primjer AVL stabala

Prednosti AVL stabala

Pitanja i odgovori

Sažmite ovu objavu uz:

Prijavite se za bilten

Što su AVL stabla?

Kako radi AVL stablo?

Faktor ravnoteže u AVL stablima

Svojstva faktora ravnoteže

AVL rotacije

Lijevo – Lijeva rotacija

Desno – Desna rotacija

Rotacija desno – lijevo

Rotacija lijevo – desno

Umetanje u AVL stabla

POVEZANI ČLANCI

Brisanje u AVL stablima

C++ Primjer AVL stabala

Prednosti AVL stabala

Pitanja i odgovori

Sažmite ovu objavu uz:

Prijavite se za bilten