A statistic related to trees and words on a finite alphabet

Abstract

Let [Formula: see text] denote the set of trees on [Formula: see text] labeled vertices. In this paper, we consider the statistic on [Formula: see text] which records the number of vertices of odd degree and count trees having a prescribed number of such vertices. In doing so, we make use of a well-known bijection between members of [Formula: see text] and the set of [Formula: see text]-ary words of length [Formula: see text]. In fact, we are able to study a generalization mod [Formula: see text] of the aforementioned statistic for all [Formula: see text]. In the cases when [Formula: see text] and [Formula: see text], further enumerative results for both words and trees may be given.