  We study tree approximations to classical two-body partition functions on
sparse and loopy graphs via the Brydges-Kennedy-Abdessalam-Rivasseau forest
expansion. We show that for sparse graphs (with large cycles), the partition
function above a certain temperature $T^*$ can be approximated by a graph
polynomial expansion over forests of the interaction graph. Within this "forest
phase", we show that the approximation can be written in terms of a reference
tree $\mathcal T$ on the interaction graph, with corrections due to cycles.
From this point of view, this implies that high-temperature models are easy to
solve on sparse graphs, as one can evaluate the partition function using belief
propagation. We also show that there exists a high- and low-temperature regime,
in which $\mathcal T$ can be obtained via a maximal spanning tree algorithm on
a (given) weighted graph. We study the algebra of these corrections and provide
first- and second-order approximation to the tree Ansatz, and give explicit
examples for the first-order approximation.
