  Using the theory of negative association for measures and the notion of
random weak limits of sparse graphs, we establish the validity of the cavity
method for counting spanning subgraphs subject to local constraints in
asymptotically tree-like graphs. Specifically, the corresponding free entropy
density is shown to converge along any sequence of graphs whose random weak
limit is a tree, and the limit is directly expressed in terms of the unique
solution to a limiting cavity equation. On a Galton-Watson tree, the latter
simplifies into a recursive distributional equation which can be solved
explicitely. As an illustration, we provide an explicit-limit formula for the
$b-$matching number of an Erd\H{o}s-R\'enyi random graph with fixed average
degree and diverging size, for any $b\in\mathbb N$.
