Ladder variables, internal structure of Galton-Watson trees and finite branching random walks.

In this paper, we consider Galton-Watson trees conditioned by size. We show that the number of k-ancestors (ancestors that have k children) of a node u is (almost) proportional to its depth. The k, j-ancestors are also studied. The methods rely on the study of ladder variables on an associated random walk. We also give an application to finite branching random walks.