The graph, K5 has 125 different spanning trees, which I beleive fit into three different non-isomorphic classes of spanning trees. However, I'm at odds as to how to figure out how many are in each class.
The trees I've found are, according to vertex degree:
(4, 1, 1, 1, 1) (3, 2, 1, 1, 1) (2, 1, 2, 1, 2)
How do I found out how many there are in each, to add up to 125?
The first one you have listed is by far the easiest, we just pick the vertex we wish to be of degree 4 (out of the 5 choices) and everything else is determined. So there are 5 of the first type.
The next easiest class for me is the second, we can first choose the vertex we wish to be of degree 3 (of 5 choices) then of the remaining 4 vertices we pick the one we wish to be of degree 2. Finally we have to decide which of the last 3 vertices is connected to the degree 2 one (equivalently which two are connected to the degree 4 one) there are 3 choices and hence $5\cdot 4 \cdot 3 = 60$ isomorphic trees.
I'll leave the derivation of the last class up to you, it does require maybe a little more care to ensure you do not over count.