Counting Forests by Descents and Leaves

Abstract

A descent of a rooted tree with totally ordered vertices is a vertex that is greater than at least one of its children. A leaf is a vertex with no children. We show that the number of forests of rooted trees on a given vertex set with $i+1$ leaves and $j$ descents is equal to the number with $j+1$ leaves and $i$ descents. We do this by finding a functional equation for the corresponding exponential generating function that shows that it is symmetric.