Consider a branching process with generation sizes Zn satisfying Z0 = 1 and P(Z1 = 0) = 0. Pick two individuals at random (with replacement) from the nth generation and let L be the index of the generation which contains their most recent common ancestor.

Show that:

P(L = r) = E[ (Zr)^-1 ] - E[ (Z(r+1))^-1 ] for 0 <= r < n.

What can be said if P(Z1 = 0) > 0?

Any help would be appreciated!