Why balanced trees?

Binary tree: $O(n)$ for insert, delete and search.

Some better trees:

• AVL Trees
• B-trees
• Splay trees
• Weight …

AVL Trees

Similar to binary search tree. But also different in a few ways:

• The height of an AVL tree is $O(log n)$.
• Each internal node has a balance property equal to -1, 0, 1.
• Balance value = height of the left subtree - height of the right sub tree.

Balance Property

Have to store the height of sub-trees.

-1 - Right heavy, +1 - Left heavy, 0 - Balanced.

• Same as BST

Insertion

After insert, if a node’s balance factor is not -1, 0 or +1, we have to do one single rotation to fix it.

If there is a “bent”, then we have to do double rotation

for (X = parent(Z); X != null; X = parent(Z)) { // Loop (possibly up to the root)
// BalanceFactor(X) has to be updated:
if (Z == right_child(X)) { // The right subtree increases
if (BalanceFactor(X) > 0) { // X is right-heavy
// ===> the temporary BalanceFactor(X) == +2
// ===> rebalancing is required.
G = parent(X); // Save parent of X around rotations
if (BalanceFactor(Z) < 0)      // Right Left Case     (see figure 5)
N = rotate_RightLeft(X, Z); // Double rotation: Right(Z) then Left(X)
else                           // Right Right Case    (see figure 4)
N = rotate_Left(X, Z);     // Single rotation Left(X)
} else {
if (BalanceFactor(X) < 0) {
BalanceFactor(X) = 0; // Z’s height increase is absorbed at X.
break; // Leave the loop
}
BalanceFactor(X) = +1;
Z = X; // Height(Z) increases by 1
continue;
}
} else { // Z == left_child(X): the left subtree increases
if (BalanceFactor(X) < 0) { // X is left-heavy
// ===> the temporary BalanceFactor(X) == –2
// ===> rebalancing is required.
G = parent(X); // Save parent of X around rotations
if (BalanceFactor(Z) > 0)      // Left Right Case
N = rotate_LeftRight(X, Z); // Double rotation: Left(Z) then Right(X)
else                           // Left Left Case
N = rotate_Right(X, Z);    // Single rotation Right(X)
} else {
if (BalanceFactor(X) > 0) {
BalanceFactor(X) = 0; // Z’s height increase is absorbed at X.
break; // Leave the loop
}
BalanceFactor(X) = –1;
Z = X; // Height(Z) increases by 1
continue;
}
}
// After a rotation adapt parent link:
// N is the new root of the rotated subtree
// Height does not change: Height(N) == old Height(X)
parent(N) = G;
if (G != null) {
if (X == left_child(G))
left_child(G) = N;
else
right_child(G) = N;
break;
} else {
tree->root = N; // N is the new root of the total tree
break;
}
// There is no fall thru, only break; or continue;
}
// Unless loop is left via break, the height of the total tree increases by 1.

Deletion

• If the key is a leaf, delete and rebalance.

Tree Height

Maximum possible height is $log(n)$, if we have $n$ nodes.

If the height is $h$, let minsize(h) be the minimum number of nodes for an AVL tree of height $h$.

Then

minsize(0) = 0,
minsize(1) = 1,
minsize(h + 2) = 1 + minsize(h + 1) + minsize(h)

minsize(h) = fib(h+2) - 1 (why?)