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We consider a conditioned Galton-Watson tree and prove an estimate of thenumber of pairs of vertices with a given distance, or, equivalently, the numberof paths of a given length.  We give two proofs of this result, one probabilistic and the other usinggenerating functions and singularity analysis.  Moreover, the second proof yields a more general estimate for generatingfunctions, which is used to prove a conjecture by Bousquet-Melou and Jansonsaying that the vertical profile of a randomly labelled conditionedGalton-Watson tree converges in distribution, after suitable normalization, tothe density of ISE (Integrated Superbrownian Excursion).

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