  Let $F(G)$ be the number of forests of a graph $G$. Similarly let $C(G)$ be
the number of connected spanning subgraphs of a connected graph $G$. We bound
$F(G)$ and $C(G)$ for regular graphs and for graphs with fixed average degree.
Among many other things we study $f_d=\sup_{G\in \mathcal{G}_d}F(G)^{1/v(G)}$,
where $\mathcal{G}_d$ is the family of $d$--regular graphs, and $v(G)$ denotes
the number of vertices of a graph $G$. We show that $f_3=2^{3/2}$, and if
$(G_n)_n$ is a sequence of $3$--regular graphs with length of the shortest
cycle tending to infinity, then $\lim_{n\to \infty}F(G_n)^{1/v(G_n)}=2^{3/2}$.
We also improve on the previous best bounds on $f_d$ for $4\leq d\leq 9$.
