Signed tree associahedra

Spines T a tree on a signed ground set V = V− t V. Spine on T = directed and labeled tree S such that • the labels of the nodes of S form a partition of the signed ground set V, • at a node labeled by U = U− t U, the source label sets of the incoming arcs are subsets of distinct connected components of TrU−, and the sink label sets of the outgoing arcs are subsets of distinct connected components of Tr U. Spine poset S(T) = poset of arc contractions on signed spines of T. Prop. The spine poset S(T) is a pure graded poset of rank |V|. Signed nested complex = simplicial complex N (T) = {N(S) | S ∈ S(T)}, where N(S) = collection of source sets of S. Exm. V− = {1, 3, 4, 5} V = {0, 2, 6, 7, 8, 9}