Vertices of Degree k in Random Unlabeled Trees

Let H n be the class of vertex-rooted unlabeled trees with n vertices, and denote by H n a tree that is drawn uniformly at random from this set. In this work we study the number deg k ( H n ) of vertices of degree k in H n . In particular, for k = O ( ( log n log log n ) 1 / 2 ) we show exponential-type bounds for the probability that deg k ( H n ) 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. 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.