To see the forest for the trees: On the infinite divisibility of unlabeled forests

TL;DR

This paper investigates the probabilistic properties of unlabeled forests, demonstrating their convergence to a shifted compound Poisson distribution and exploring related asymptotic proportions and limit theories.

Contribution

It introduces a probabilistic framework for unlabeled forests, showing convergence to a shifted compound Poisson and linking existing results through Levy process theory.

Findings

Number of trees in a random forest converges to a shifted compound Poisson.

Asymptotic proportion of forests that are trees is characterized.

Results apply to enumeration of weighted integer partitions and Levy processes.

Abstract

Inspired by Stufler's recent probabilistic proof of Otter's asymptotic number of unlabeled trees, we revisit work of Palmer and Schwenk, and study unlabeled forests from a probabilistic point of view. We show that the number of trees in a random forest converges, with all of its moments, to a shifted compound Poisson. We also find the asymptotic proportion of forests that are trees. The key fact is that the number of trees $t_n$ and forests $f_n$ are related by a L\'evy process. As such, the results by Palmer and Schwenk follow by an earlier and far-reaching limit theory by Hawkes and Jenkins. We also show how this limit theory implies results by Schwenk and by Meir and Moon, related to degrees in large random trees. Our arguments apply, more generally, to the enumeration of sub-exponentially weighted integer partitions, or, in fact, any setting where the underlying L\'evy process follows the one big jump principle.