Distances Between Pairs of Vertices and Vertical Profile in Conditioned Galton-Watson Trees
We consider a conditioned Galton-Watson tree and prove an estimate of the number of pairs of vertices with a given distance, or, equivalently, the number of paths of a given length. We give two proofs of this result, one probabilistic and the other using generating functions and singularity analysis. Moreover, the latter proof yields a more general estimate for generating functions, which is used to prove a conjecture by Bousquet-Melou and Janson (Bousquet-Melou and Janson, Ann Appl Probab 16 (2006) 1597-1632), saying that the vertical profile of a randomly labelled conditioned Galton-Watson tree converges in distribution, after suitable normalization, to the density of ISE (Integrated Superbrownian Excursion). (C) 2010 Wiley Periodicals, Inc. Random Struct. Alg., 38, 381-395, 2011
vertical profile random galton-watson tree paths of given length probabilistic analysis of algorithms branching process