  The number of spanning trees in the giant component of the random graph
$\G(n, c/n)$ ($c>1$) grows like $\exp\big\{m\big(f(c)+o(1)\big)\big\}$ as
$n\to\infty$, where $m$ is the number of vertices in the giant component. The
function $f$ is not known explicitly, but we show that it is strictly
increasing and infinitely differentiable. Moreover, we give an explicit lower
bound on $f'(c)$. A key lemma is the following. Let $\PGW(\lambda)$ denote a
Galton-Watson tree having Poisson offspring distribution with parameter
$\lambda$. Suppose that $\lambda^*>\lambda>1$. We show that $\PGW(\lambda^*)$
conditioned to survive forever stochastically dominates $\PGW(\lambda)$
conditioned to survive forever.
