Distances between pairs of vertices and vertical profile in conditioned Galton--Watson trees

Details Distances between pairs of vertices and vertical profile in conditioned Galton--Watson trees Distances between pairs of vertices and vertical profile in conditioned Galton--Watson trees

2008-12-17

arxiv.org/abs/0812.3326v1

Mathematics

Probability

16 pages

Scientific paper

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 second proof yields a more general estimate for generating functions, which is used to prove a conjecture by Bousquet-Melou and Janson 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).

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