A refined enumeration of hex trees and related polynomials

A hex tree is an ordered tree of which each vertex has updegree 00, 11, or 22, and an edge from a vertex of updegree 11 is either left, median, or right. We present a refined enumeration of symmetric hex trees via a generalized binomial transform. It turns out that the refinement has a natural combinatorial interpretation by means of supertrees. We describe a bijection between symmetric hex trees and a certain class of supertrees. Some algebraic properties of the polynomials obtained in this procedure are also studied.