On the Profile of Random Forests

We consider the set F (n,N) of random forests consisting of n vertices and N rooted trees which can be viewed as realizations of Galton-Watson branching processes with N initial particles and conditioned to have total progeny n. Such forests consist of simply generated trees according to Meir and Moon [20] and therefore they can easily be described by generating functions: Let b(z) = ∑ n≥0 bn,Nz n denote the generating function for those forests. Then we have b(z) = a(z) with a(z) = zφ(a(z)). Here a(z) is the generating function for a single tree and φ(t) = ∑ n≥0 φnt n is the generating function of an arbitrary sequence (φk)k≥0 of nonnegative numbers with φ0 > 0. In this setting bn,N can be viewed as the number of forests in F (n,N), weighted according to the probability on F (n,N), i.e., to each forest F is assigned a weight