Given a complete graph $K_n$ and some vertex $v\in K_n$, let $c_n(k)$ be the numbers of spanning trees of $K_n$ with $\deg(v) = k$. Prove that
$$c_n(k) = \frac{k(n-1)}{n-1-k}c_n(k+1)$$
This is used in particular as an alternate proof for Cayley's formula, so I'm trying to avoid using both Cayley's formula and Prüfer codes. I read about the summation version of Cayley's formula:
Cayley's formula summation form
Which seems relevant, as to go from this to Cayley's formula I will probably sum over all $k$ which will give some similar expressions, but I am not sure exactly how to relate to it.
The only real direction I had so far is trying to build graphs with $\deg(v)=k-1$ given a graph with $\deg(v)=k$, but the number of such graphs depends on the particular graph we are given, and while counting the variations is possible, it gives a direct computation of $c_n(k)$ more then a recursive formula for $c_n(k-1)$, so I am not exactly sure how to proceed.