on counting of special case of trees on a graph

I'll answer a question raised in the comments:

Problem: Count the number of induced trees of size $k$.

According to this paper by Erdös, Saks and Sos, it is NP-complete to decide given a graph $G$ and an integer $k$, if $G$ contains an induced tree of size $k$. So, it's probably pretty damn hard to count them. Apparently, it remains NP-complete even for bipartite graphs.

Actually, the argument is pretty simple so I'll include it here. Given a graph $H$ and an integer $k$, it is well-known that the problem of deciding if $H$ has an independent set of size $k$ is NP-complete. Suppose that $H$ has $n$ vertices. Let $G$ be the graph obtained from $H$ by first adding a disjoint copy of $P_n$ (a path on $n$ vertices), and then connecting one end of $P_n$ to all the vertices in $H$. Clearly, $H$ has an independent set of size $k$ if and only if $G$ contains an induced tree of size $n+k$.