  Discrete statistical models supported on labelled event trees can be
specified using so-called interpolating polynomials which are generalizations
of generating functions. These admit a nested representation. A new algorithm
exploits the primary decomposition of monomial ideals associated with an
interpolating polynomial to quickly compute all nested representations of that
polynomial. It hereby determines an important subclass of all trees
representing the same statistical model. To illustrate this method we analyze
the full polynomial equivalence class of a staged tree representing the best
fitting model inferred from a real-world dataset.
