The distance to the border of a random tree

Given a Galton-Watson process conditioned to have total progeny equal to $n$, we study the asymptotic probability that this conditioned Galton-Watson process has distance to the border bigger or equal than $k$, as the number of nodes $n \rightarrow \infty$. A problem which is akin to this one was solved by R\'enyi and Szekeres for Cayley trees, de Bruijn, Knuth, and Rice for plane trees and Flajolet, Gao, Odlyzko, and Richmond for binary trees. The distance to the border is dual, in a certain sense, to the height. The first of these distances is the minimum of the distances from the root to the leaves. The second is the maximum of the distances from the root to the leaves. These are two extreme complementary cases.