标题： 看森林而不看树木：关于无标签森林的无限可分性 显示英文标题

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

摘要： 受Stufler最近对Otter渐近无标树数量的概率证明的启发，我们重新研究了Palmer和Schwenk的工作，并从概率的角度研究了无标森林。 我们证明了一个随机森林中的树的数量在所有矩的情况下都收敛到一个偏移的复合泊松分布。 我们还找到了森林中为树的渐近比例。 关键事实是，树的数量$t_n$和森林的数量$f_n$由一个Lévy过程相关联。 因此，Palmer和Schwenk的结果可以通过Hawkes和Jenkins早期且影响深远的极限理论得出。 我们还展示了这个极限理论如何推导出与大随机树中度数相关的Schwenk以及Meir和Moon的结果。 我们的论证更普遍地适用于次指数加权整数分拆的计数，或者实际上，任何底层Lévy过程遵循一次大跳跃原理的情况。 显示英文摘要

摘要： 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.