Hello. After flipping through a few textbooks on birth-death processes, I can't seem to find anything about genealogical distribution of survivors (conditioned on non-extinction). What I am looking for is a statement roughly of the type, "in generation n, there is probability at least P(j,k,n) that descendants have survived from at least k distinct members of generation j".
I'm interested in the critical case of the Galton-Watson process, where the number of descendents is i.i.d. with mean 1. If necessary, assume the distribution is Bin(r,1/r).
Also, my sample question about P(j,k,n) is just the easiest thing I could think to write down. But I would like to know how likely it is for there to exist a very uniformly distributed subpopulation of size roughly n in generation n.