$G$ is an element of FBRT (full binary rooted trees), $v(G)$ = total vertices in $G$, and $e(G)$ = total edges in $G$. I know logically that this is true, but I'm not sure how to prove it using structural induction.
Edit: I already figured out a way to prove this part, but I also need to know what goes wrong when you try to prove this for EBRTs (extended binary rooted trees)? I feel like this formula should also apply to EBRTs, but this is what it says on my worksheet, which I don't get why.
For the base case, pick the simplest tree you can imagine (i.e. one vertex, no edges) and verify that the formula works.
Proceeding inductively, we take some tree $T$ and suppose it is true for all smaller trees. If you pick a vertex $v \in T$ and let $T' = T \setminus v$, then we know the claim holds for $T'$ by assumption. Use this to deduce that it holds for $T$ too.