  We present a link-by-link rule-based method for constructing all members of
the ensemble of spanning trees for any recursively generated, finitely
articulated graph, such as the DGM net. The recursions allow for many
large-scale properties of the ensemble of spanning trees to be analytically
solved exactly. We show how a judicious application of the prescribed growth
rules selects for certain subsets of the spanning trees with particular desired
properties (small-world or extended diameter, degree distribution, etc.), and
thus provides solutions to several optimization problems. The analysis of
spanning trees enhances the usefulness of recursive graphs as sophisticated
models for everyday life complex networks.
