  In 1986, Janson showed that the number of edges in the union of $k$ random
spanning trees in the complete graph $K_n$ is a shifted Poisson distribution.
Using results from the theory of electrical networks, we provide a new proof of
this result, and we obtain an explicit rate of convergence. This rate of
convergence allows us to show a new upper tail bound on the number of trees in
$G(n,p)$, for $p$ a constant not depending on $n$. The number of edges in the
union of $k$ random trees is related to moments of the number of spanning trees
in $G(n, p)$.
  As an application, we prove the law of the iterated logarithm for the number
of spanning trees in $G(n,p)$. More precisely, consider the infinite random
graph $G(\mathbb{N}, p)$, with vertex set $\mathbb{N}$ and where each edge
appears independently with constant probability $p$. By restricting to $\{1, 2,
\dotsc, n\}$, we obtain a series of nested Erd\"{o}s-R\'{e}yni random graphs
$G(n,p)$. We show that a scaled version of the number of spanning trees
satisfies the law of the iterated logarithm.
