Induction proof $2^{n+1}-1$ when n is 50

Question

There are 50 of YES or NO questions. Supposed store them into a binary tree. Each path from root to leaf implies a possible answer to the questions.

The number of vertices for $n$ question is

$2^{n+1}-1$

How to prove $2^{51}-1$ vertices in the case?

Answer 1

If $n$ questions give a total of $2^{n+1}-1$ nodes, and moreover have $2^n$ child-less nodes, then for $n+1$ questions, each child-less node has two children, or $2^{n+1}$ additional nodes.

So:

$$\text{nodes}(n+1)=\text{nodes}(n)+2n=2^{n+1}-1+2^{n+1}$$ $$=2^{n+2}-1$$