AVL Tree - Highest number of nodes between LST and RST

Question

I have an AVL Tree, n nodes in it, what's the maximum difference of nodes that can be from left sub root tree and right sub root tree?

So I'm not quite sure but first of all let's fill all the right sub tree with nodes, so we have n/2 + 1 nodes filled, we're left with n - n/2 + 1 nodes to full in the left sub tree, and I want to insert the minimum nodes now there so the height won't be higher than 1. so I believe I need to add about logn - 1 nodes, like a chain (always to the right child or always to the left child).

second idea, maybe build two chains of nodes, one in the right sub-tree and one in the left sub-tree so the height won't exceed 1 between the sub trees. but this won't be the maximum difference i guess

Any ideas?

score 3 · Accepted Answer · answered Sep 11 '16 at 12:54

3

The left and right subtrees differ by at most 1, by definition. Say the larger one has height h + 1, and the smaller one has height only h.

For the larger one, the most nodes that can be in a tree of height h + 1 is ~ 2^h (the case of a full tree).

For the smaller one, the number of nodes is known to be 1 / √5 ((1 + √5) / 2)^{h + 3} - 1.

So, the answer is 2^h - 1 / √5 ((1 + √5) / 2)^{h + 3} + 1, subject to 2^h + 1 / √5 ((1 + √5) / 2)^{h + 3} - 1 = n.

For large h and n, 2^h + 1 / √5 ((1 + √5) / 2)^{h + 3} - 1 ~ 2^h, so 2^h ~ n.

Surprisingly (at least for me), the difference is Ω(n).

answered Sep 11 '16 at 12:54

Ami Tavory

74,578
11
141
185

Was trying to figure out by myself the number of nodes in the smaller subtree. Thanks for the reference, much more complicated than what I expected. – A. Sarid Sep 11 '16 at 12:58
@A.Sarid :-) Good for you for trying to develop it. – Ami Tavory Sep 11 '16 at 13:01

AVL Tree - Highest number of nodes between LST and RST

1 Answers1