@article {IOPORT.06004151,
    author       = {Panagiotou, Konstantinos and Sinha, Makrand},
    title        = {Vertices of degree $k$ in random unlabeled trees.},
    year         = {2012},
    journal      = {Journal of Graph Theory},
    volume       = {69},
    number       = {1-2},
    issn         = {0364-9024},
    pages        = {114-130},
    publisher    = {John Wiley \& Sons, New York, NY},
    doi          = {10.1002/jgt.20567},
    abstract     = {Summary: Let ${\Cal H}_n$ 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 {\it M. Drmota} and {\it B. Gittenberger} in [J. Graph Theory 31, No. 3, 227--253 (1999; Zbl 0929.05019)], 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 $k={\Cal O}((\log n/(\log\log n))^{1/2})$ 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, see e.g. the work [{\it N. Bernasconi}, {\it K. Panagiotou} and {\it A. Steger}, ``On properties of random dissections and triangulations,'' Proceedings of the nineteenth annual ACM-SIAM symposium on discrete algorithms, San Francisco, CA, 2008. New York, NY: Association for Computing Machinery (ACM); Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM). 132--141 (2008; Zbl 1192.05141) and ``On the degree sequences of random outerplanar and series-parallel graphs,'' Goel, Ashish (ed.) et al., Approximation, randomization and combinatorial optimization. Algorithms and techniques. 11th international workshop, APPROX 2008, and 12th international workshop, RANDOM 2008, Boston, MA, USA, 2008. Proceedings. Berlin: Springer. Lecture Notes in Computer Science 5171, 303--316 (2008; Zbl 1159.68635)]. 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.},
    identifier   = {06004151},
}