  Let $\ell$ be a rational prime. We show that an analogue of a conjecture of
Greenberg in graph theory holds true. More precisely, we show that when $n$ is
sufficiently large, the $\ell$-adic valuation of the number of spanning trees
at the $n$th layer of a $\mathbb{Z}_{\ell}^{d}$-tower of graphs is given by a
polynomial in $\ell^{n}$ and $n$ with rational coefficients of total degree at
most $d$ and of degree in $n$ at most one.
