I have to prove the following claim, given the tree $T=(V,E)$, $|V|=n\geq2$: $|V_1| = 2 + \sum (j-2)|V_j|$
where the sum is from $j=3$ up to the highest degree, and $V_i = \{ x \in V \mid \deg(x) = i\}.$
This was a bonus question given by my professor. We were sitting on this question for hours and have no idea how to prove it.
Can someone help out with a hint?
Thanks!