Question

What is the minimum and maximum number of nodes in a balanced avl tree of height 5

Answer 1

Answer:

Minimum number of nodes in an AVL Tree with given height

Given the height of an AVL tree ‘h’, the task is to find the minimum number of nodes the tree can have.

Examples :

Input : H = 0

Output : N = 1

Only '1' node is possible if the height

of the tree is '0' which is the root node.

Input : H = 3

Output : N = 7

Recommended: Please try your approach on {IDE} first, before moving on to the solution.

Recursive Approach : In an AVL tree, we have to maintain the height balance property, i.e. difference in the height of the left and the right subtrees can not be other than -1, 0 or 1 for each node.

We will try to create a recurrence relation to find minimum number of nodes for a given height, n(h).

For height = 0, we can only have a single node in an AVL tree, i.e. n(0) = 1

For height = 1, we can have a minimum of two nodes in an AVL tree, i.e. n(1) = 2

Now for any height ‘h’, root will have two subtrees (left and right). Out of which one has to be of height h-1 and other of h-2. [root node excluded]

So, n(h) = 1 + n(h-1) + n(h-2) is the required recurrence relation for h>=2 [1 is added for the root node]