For a complete graph $K_n$ on n vertices with weighted edges,
define the weight of a spanning tree (more generally, spanning forest) as the
product of edge weights involved. Define the tree weight (forest weight) of
$K_n$ as the total weight of all spanning trees (forests). The uniform edge
weight distribution is shown to maximize the tree weight, and an explicit bound
on the tree weight is formulated in terms of the overall variance of edge
weights as well as the variance of the sum of edge weights over nodes. An
application to sparse random graphs leads to a bound for the relative risk of
observing a spanning tree in a well-defined neighborhood of the uniform
distribution. An analogous result shows that, for each positive integer
k, the weight of all forests with k rooted trees is also
maximized under the uniform distribution. A key ingredient for the latter
result is a formula for the weight of forests of k rooted trees that
generalizes Maxwell's rule for spanning trees. Our formulas also enable us to
show that the number of trees in a random rooted forest is intrinsically
divisible, that is, representable as a sum of n independent binary
random variables $\varepsilon_j$, with parameter $(1 + \lambda_j)^{-1},
\lambda$'s being the eigenvalues of the Kirchhoff matrix. This is directly
analogous to the properties of the number of blocks in a random set partition,
of the size of the random matching set and of the number of
leaves in a random tree.