Vertices of degree k in random unlabeled trees

Let ℋ n be the class of unlabeled trees with n vertices, and denote by H n a tree that is drawn uniformly at random from this set. The asymptotic behavior of the random variable deg k ( H n ) that counts vertices of degree k in H n was studied, among others, by Drmota and Gittenberger in [J Graph Theory 31(3) (1999), 227–253], 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 H n . In particular, for k = ((log n /(loglog 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, see e.g. the work [Bernasconi et al., SODA '08: Proceedings of the 19th Annual ACM‐SIAM Symposium on Discrete Algorithms, Society for Industrial and Applied Mathematics, Philadelphia, PA, 2008, pp. 132–141; Bernasconi et al., Proceedings of the 11th International Workshop, APPROX 2008, and 12th International Workshop, RANDOM 2008 on Approximation, Randomization and Combinatorial Optimization, Springer, Berlin, 2008, pp. 303–316] 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. © 2011 Wiley Periodicals, Inc. J Graph Theory. 69:114‐130, 2012