In this paper we give a complete characterization of the statistical
equivalence classes of CEGs and of staged trees. We are able to show that all
graphical representations of the same model share a common polynomial
description. Then, simple transformations on that polynomial enable us to
traverse the corresponding class of graphs. We illustrate our results with a
real analysis of the implicit dependence relationships within a previously
studied dataset.Comment: 18 pages, 4 figure