Question: A very balanced rooted plane binary tree (VBRPBT for short) is defined to be either An empty tree, or A single node, attached

A very balanced rooted plane binary tree (VBRPBT for short) is defined to be either

• An empty tree, or

• A single node, attached to two subtrees L and R, both of which are VBRPBTs, where the difference between the number of nodes in L and the number of nodes in R is at most 1. (As a formula, |s(L) − s(R)| ≤ 1.)

(a) How many VBRPBTs with 10 nodes are there? (There is 1 with 1 node, 2 with 2 nodes, 1 with 3 nodes, and 4 with 4 nodes, as drawn below.)

(b) Prove that, for any number n, there are 2k VBRPBTs for some k (which you don’t have to figure out explicitly). (You probably want to use induction. Hopefully in doing the previous part you have figured out a recursive formula (with two recursive cases) for the numberof VBRPTs of size n.)

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