Shellability of Complexes of Trees

A very interesting abstract simplicial complex T (k) n has faces in bijection with the trees with at most n interior vertices, all of which have degrees at least k+2 and are congruent to 2 mod k, and whose leaves are labelled by the distinct integers in [0, 1, ..., m], where m+1 :=nk+2 is the number of leaves (n 0, k 1). Thus the facets of T n correspond to the leaf-labelled trees with n interior vertices of degree exactly k+2, while the vertices of the complex correspond to the trees with exactly one interior edge, and two internal nodes of degrees kl+2 and k(n&l )+2, with 1 l n&1. The partial order on these trees that is induced by contraction of interior edges corresponds to inclusion relation between faces of the complex T (k) n . The complex T (k) n has n&1 i=1 ( m ki+1) vertices. Its dimension is n&2. For example, for n=3 and k=2 we obtain a 1-dimensional simplicial complex (i.e., a graph) with (3)+( 7 5)=56 vertices corresponding to graphs with one interior edge as depicted on the left of our picture, and 2 ( 8 2)( 6 3) facets (graph edges) as depicted on the right. Article No. TA972844