Given a weighted simple undirected connected graph $G = (V, E, w:E to mathbb{R})$, let $tau(G)$ be the set of all its spanning trees. Is there an efficient algorithm to determine or estimate with high probability the number

$W = sum_{mathcal{T} in tau(G)} sum_{e in E(mathcal{T})} w(e)$

without actually enumerating all the spanning trees? Are there asymptotic estimates on this number?