Finding the minimum and maximum height in a AVL tree, given a number of nodes?

Question

Is there a formula to calculate what the maximum and minimum height for an AVL tree, given a certain number of nodes?

For example:
Textbook question

Answer 1

Lets assume the number of nodes is n

Trying to find out the minimum height of an AVL tree would be the same as trying to make the tree complete i.e. fill all the possible nodes at each level and then move to the next level.

So at each level the number of eligible nodes increases by 2^(h-1) where h is the height of the tree.

So at h=1, nodes(1) = 2^(1-1) = 1 node

for h=2, nodes(2) = nodes(1)+2^(2-1) = 3 nodes

for h=3, nodes(3) = nodes(2)+2^(3-1) = 7 nodes

so just find the smallest h, for which nodes(h) is greater than the given number of nodes n.

Now for the problem of maximum height of an AVL tree:-

lets assume that the AVL tree is of height h, F(h) being the number of nodes in the AVL tree,

for its height to be maximum lets assume that its left subtree FL and right subtree FR have a difference in height of 1(as it satisfies the AVL property).

Now assuming FL is a tree with height h-1 and FR be a tree with height h-2.

now the number of nodes in

F(h)=F(h-1)+F(h-2)+1 (Eq 1)

Adding 1 on both sides :

F(h)+1=(F(h-1)+1)+ (F(h-2)+1) (Eq 2)

So we have reduced the maximum height problem to a Fibonacci sequence. And these trees F(h) are called Fibonacci Trees.

So, F(1)=1 and F(2)=2

so in order to get the maximum height just find the index of the the number in the fibonacci sequence which is less than or equal to n.

So applying (Eq 1)

F(3)= F(2) + F(1)+ 1=4, so if n is between 2 and 4 tree will have height 3.

F(4)= F(3)+ F(2)+ 1 = 7, similarly if n is between 4 and 7 tree will have height 4.

and so on.