I am asked how many trees are in an n-cycle. It doesn't specify that these trees must be spanning trees, just trees. I know the number of trees on n labelled vertices is nexp(n-2), but this isn't necessarily on vertices that form a cycle.
Any help would be appreciated, thanks!
Trees are connected graphs that do not contain a cycle. Cycles do not properly contain other cycles, so any proper subgraph of a cycle does not contain a cycle. The goal is therefore to count connected proper subgraphs. For example, subgraphs consisting of two adjacent verices with their adjoining edge are trees; there are $n$ of those. Provided $n>3$, induced subgraphs with three adjacent vertices along the cycle are connected and contain no cycle, and there are $n-1$ of these; etc.