Let $G$ be a connected simple graph, and identify two vertices $s$ and $t$. Let $\tau(G)$ be the number of spanning trees of $G$, and let $f(G)$ be the number of spanning forests of $G$ with $2$ components such that $s$ and $t$ are in different components.

Starting from a spanning tree of $G$, one can delete any of its $n-1$ edges to form a spanning forest with $2$ components. This gives the upper bound $f(G) \leq (n-1)\, \tau(G)$, which holds with equality if $G$ is simply an $st$-path.

Are any other upper bounds known for $f(G)$ in terms of $\tau(G)$ and other graph parameters?

I suspect another bound can be derived from the answers about effective resistance here, but I'm having trouble working one out.