To see the forest for the trees: On the infinite divisibility of unlabeled forests
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arXiv
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| Format: | Preprint |
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2025
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| author | Bassan, Michal Donderwinkel, Serte Kolesnik, Brett |
| author_facet | Bassan, Michal Donderwinkel, Serte Kolesnik, Brett |
| contents | 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 Lévy 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 Lévy process follows the one big jump principle. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2507_16650 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | To see the forest for the trees: On the infinite divisibility of unlabeled forests Bassan, Michal Donderwinkel, Serte Kolesnik, Brett Probability Combinatorics 05A16, 05A17, 05C05, 05C30, 60E07 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 Lévy 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 Lévy process follows the one big jump principle. |
| title | To see the forest for the trees: On the infinite divisibility of unlabeled forests |
| topic | Probability Combinatorics 05A16, 05A17, 05C05, 05C30, 60E07 |
| url | https://arxiv.org/abs/2507.16650 |