Number of pairwise non-isomorphic spanning trees of the wheel $W_n$, with restrictions

Question

I recently encountered this problem. Frankly I'm stuck; would be nice for some help. Here it is:

Let $N,k$ be positive integers. By $p_k(N)$ we denote the number of integer partitions of $N$ with exactly $k$ parts; that is, the number of ways to write $N$ as a sum of exactly $k$ positive integers, where the order of summands does not matter. What is the number of pairwise non-isomorphic spanning trees of $W_n$ of maximum degree at least 4 and with at most 1 vertex of degree 3? Write this number using terms of the form $p_k(N)$.

Answer 1

A beginning:

You should give your exact definition of a wheel $W_n$, as the notation is inconsistent in the literature.

It is required that the spanning tree $T$ has a vertex of degree $\geq4$, and this has to be the center of the wheel. All other vertices of $T$ have degree $\leq3$, at most one of them degree $3$. The latter can happen only in the following way: One spike emanating from the center bifurcates at its end.

Now draw a few pictures in order to get an overwiew of what is possible. Note that the cyclic arrangement of the spikes is irrelevant when it comes to isomorphy.