  We calculate exponential growth constants $\phi$ and $\sigma$ describing the
asymptotic behavior of spanning forests and connected spanning subgraphs on
strip graphs, with arbitrarily great length, of several two-dimensional
lattices, including square, triangular, honeycomb, and certain heteropolygonal
Archimedean lattices. By studying the limiting values as the strip widths get
large, we infer lower and upper bounds on these exponential growth constants
for the respective infinite lattices. Since our lower and upper bounds are
quite close to each other, we can infer very accurate approximate values for
these exponential growth constants, with fractional uncertainties ranging from
$O(10^{-4})$ to $O(10^{-2})$. We show that $\phi$ and $\sigma$, are
monotonically increasing functions of vertex degree for these lattices.
