Vertices of Degree k in Random Unlabeled Trees

Let Hn be the class of unlabeled trees with n vertices, and denote by Hn a tree that is drawn uniformly at random from this set. The asymptotic behavior of the random variable degk(Hn) that counts vertices of degree k in Hn was studied, among others, by Drmota and Gittenberger in [3], who showed that this quantity satisfies a central limit theorem. This result provides a very precise characterization of the “central region” of the distribution, but does not give any non-trivial information about its tails. In this work we study further the number of vertices of degree k in Hn. In particular, for k = O(( logn log logn )) we show exponential-type bounds for the probability that degk(Hn) deviates from its expectation. On the technical side, our proofs are based on the analysis of a randomized algorithm that generates unlabeled trees in the so-called Boltzmann model. The analysis of such algorithms is quite well-understood for classes of labeled graphs, see e.g. the work [1, 2] by Bernasconi, the first author, and Steger. Comparable algorithms for unlabeled classes are unfortunately much more complex. We demonstrate in this work that they can be analyzed very precisely for classes of unlabeled graphs as well.