AVL Trees

AVL Node Structure The AVL node structure follows the

same structure as the binary search tree, with the addition of a term to store the balance factor.

Node { int data; Node *left; Node *right; int balanceFactor;}

AVL Insert Now that we have seen how to

balance a tree, we are ready to look at the algorithms.

The search and retrieval algorithms are the same as for any binary tree.

However, because the AVL tree is a special case of a binary search tree, you will want to use an inorder traversal method.

AVL Insert As with the BST, all inserts take place at

a leaf (or leaf-like) node. To find the appropriate leaf node, we

follow the path from the root, going left when the new data node’s key is less than the root node’s key and right when it’s greater.

Once we have found the leaf, we connect the new node to the leaf and begin to back out of the tree.

AVL Insert It is at this point that the AVL

insert differs from the BST insert. As we back out of the tree, we

constantly check the balance of each node.

When we find that a node is out of balance, we balance it and then continue up the tree.

Note that not all inserts will produce an out-of-balance tree.

AVL Delete The delete logic is similar to the

BST delete logic. Again, however, we must make

sure that we include the logic to keep the tree balanced.

AVL - height 1

AVL - height 2

AVL - height 3

AVL - height 4

AVL - height 5

AVL - height 6

Counting nodes

Height 0 1 2 3 4 5 6 7 8 9 10Num Nodes 1 2 4 7 12 20 33 54 88 143 232

Relationship to Fibonacci Let N be the fewest number of

nodes in an AVL tree of height H It is straightforward to show thatN = F(H+3) - 1,where F(k) is the kth Fibonacci number

For large values of k, Fk 1

5

1 52

k

number of nodes The fewest number of nodes in an

AVL tree with height H is given by

N 15

1 52

H3 1

Solving for H if we solve this near equality for H,

we getH 1.44 log2 N

This means that the height of an AVL tree with N nodes is no more than 44% larger than the optimal height of a binary search tree with N nodes