The distribution of degrees in a large random tree

Abstract

For labeled trees, Rényi showed that the probability that an arbitrary point of a random tree has degree k approaches l/e(k−l)!. For unlabeled trees, the answer is different because the number of ways to label a given tree depends on the order of its automorphism group. Using arguments involving combinatorial enumeration and asymptotics, we evaluate the corresponding probabilities for large unlabeled trees.