A good leaf order on simplicial trees

Abstract: Using the existence of a good leaf in every simplicial tree, we order the facets of a simplicial tree in order to find combinatorial information about the Betti numbers of its facet ideal. Applications include an Eliahou-Kervaire splitting of the ideal, as well as a refinement of a recursive formula of H`a and Van Tuyl for computing the graded Betti numbers of simplicial trees.