  Working with tree graphs is always easier than with loopy ones and spanning
trees are the closest tree-like structures to a given graph. We find a
correspondence between the solutions of random K-satisfiability problem and
those of spanning trees in the associated factor graph. We introduce a modified
survey propagation algorithm which returns null edges of the factor graph and
helps us to find satisfiable spanning trees. This allows us to study
organization of satisfiable spanning trees in the space spanned by spanning
trees.
