EDIT: I think everyone understood, but I never explicitly stated that I am looking at labeled spanning trees.
Let $T$ be a tree contained in $K_n$ (the complete graph on $n$ vertices). How can one compute (or at least bound) the size of the set $ \{S \mid S \text{ a labeled spanning tree of } K_n, T \not\subseteq S \}? $
If we aren't interested in avoiding any tree, then Cayley's formula gives $n^{n-2}$ spanning trees of $K_n$. Thus, we could alternatively ask for the size of the set $ \{S \mid S \text{ a labeled spanning tree of }K_n, T \subseteq S \} $ and subtract.
More generally, I would be interested in solving the problem for avoiding a forest contained in $K_n$, but perhaps this can be deduced from the case of a tree via inclusion-exclusion.