  This paper deals with the construction of a correlation decay tree
(hypertree) for interacting systems modeled using graphs (hypergraphs) that can
be used to compute the marginal probability of any vertex of interest. Local
message passing equations have been used for some time to approximate the
marginal probabilities in graphs but it is known that these equations are
incorrect for graphs with loops. In this paper we construct, for any finite
graph and a fixed vertex, a finite tree with appropriately defined boundary
conditions so that the marginal probability on the tree at the vertex matches
that on the graph. For several interacting systems, we show using our approach
that if there is very strong spatial mixing on an infinite regular tree, then
one has strong spatial mixing for any given graph with maximum degree bounded
by that of the regular tree. Thus we identify the regular tree as the worst
case graph, in a weak sense, for the notion of strong spatial mixing.
