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 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). 
Luc Devroye and Svante Janson