I have some troubles with counting spanning trees, it seems completely abstract to me.
First one is cycle with $n$ vertices - it's just $n$, because we can move each number $n$ times like so: $1234$ $2341$ $3412$ $4123$ etc
Second graph of a tetrahedron, we can either have some vertex with degree $3$, and there are $4$ graphs like that because we simply put each number from $1$ to $4$ as this vertex and then we can have a tree with $\max(\deg(v_i)=2$ and this is where I fail.
How many isomorphic labelled trees with max degree $2$ are there? Is it $n!/2$ because $1, 2, ..., n-1,n$ is the same as $n, n-1, ..., 2, 1$?
Next, we have $2$ cycles, one with $n$ vertices and second with $m$ vertices, and we combine them so that they have:
a) $1$ common vertex, which is easy because it has just $nm$ spanning trees.
b) $1$ common edge, and here I fail, I have no idea where to even begin.