  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).
