A Refinement of Cayley's Formula for Trees

which reduce to (n + 1) for a = b = c = 1. We refine Cayley’s formula by showing that Pn(a, b, c) counts rooted forests by the number of trees and the number of proper vertices, which are vertices that are less than all of their proper descendants. Moreover, other evaluations of Pn(a, b, c) have similar interpretations for other types of trees and forests: k-ary trees, forests of ordered trees, and forests of k-colored ordered trees (which reduce to ordered trees for k = 1). We also give an interpretation of Pn(a, b, c) in terms of parking functions. This work developed from a problem posed by Alexander Postnikov at the Richard Stanley 60th Birthday Conference. He gave the following identity, which he had derived