### Abstract:

We consider the generating polynomial of the number of rooted trees on the set
\({1,2,\dots,n}\) counted by the number of descending edges (a parent with a greater label than a
child). This polynomial is an extension of the
descent generating polynomial of the set of permutations of a totally ordered \(n\)-set, known as
the Eulerian polynomial. We show how this extension shares some of the properties of the
classical one. B. Drake proved that this polynomial factors completely over the integers.
From his product formula it can be concluded that this polynomial has positive coefficients in
the \(\gamma\)-basis and we show that a formula for these coefficients can also be derived.
We discuss various combinatorial interpretations of these positive
coefficients in terms of leaf-labeled binary trees and in terms of the
Stirling permutations introduced by Gessel and Stanley. These interpretations are derived from
previous results of the author and Wachs related to the poset of weighted
partitions and the free multibracketed Lie algebra.