Denumerations of rooted trees and multisets

R\I stands for the set of positive integers, and li$, for the set of non-negative integers. If S is a set then IS( is the number of elements of S. If S is a set, then a multiset of elements of S is a mapping f of S into &, with the property that the set of s with f(s)>0 is finite. The set of all these multisets is denoted M(S). Some of our multisets are multisets of natural numbers. Number theorists might prefer the name ‘partition’ for this kind of multiset. If f EM(N), and if n = CT==, j f(j), then this f provides a way to write n as a sum of positive integers, viz. the sum off (I) one’s, f(2) two’s, f (3) three’s etc. In number theory this is called a partition of n. If V is a set, then P2(V) is the set of all 2-element subsets of V, i.e., the set