Hopf-algebraic structures of families of trees

In this paper we describe Hopf algebras which are associated with certain families of trees. These Hopf algebras originally arose in a natural fashion: one of the authors [5] was investigating data structures based on trees, which could be used to efficiently compute certain differential operators. Given data structures such as trees which can be multiplied, and which act as higherorder derivations on an algebra, one expects to find a Hopf algebra of some sort. We were pleased to find that not only was there a Hopf algebra associated with these data structures, but that it could be used to give new proofs of enumerations of such objects as rooted trees and ordered rooted trees. Previous work applying Hopf algebras to combinatorial objects (such as [10], [13] or [14]) has concerned itself with algebraic structures on polynomial algebras and on partially ordered sets, rather than on trees themselves.