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Distances between pairs of vertices and vertical profile in conditioned Galton-Watson trees. (English) Zbl 1223.05049
Summary: 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-Mélou and Janson [M. Bousquet-Mélou and S. Janson, Ann. Appl. Probab. 16, No. 3, 1597–1632 (2006; Zbl 1132.60009)], 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).

05C12 Distance in graphs
05C05 Trees
05C38 Paths and cycles
68W40 Analysis of algorithms
Zbl 1132.60009
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