Let $Z_n$ denote the population of the $n$-th generation of a Galton Watson Process, with

• $Z_0=N>1$
• $p_0 \in (0,1)$
• supercritical, i.e. the mean of descendeants is above $1$
• $(Z_n$) is conditioned on survival

Now let $K\in \mathbb{N}$. Uniformly choose an individual of the $K$-th generation, kill it and everyone sharing a blood line with it. Let $prop(K)$ denote the proportion of the size of the $K$-th generation killed in this way.

How large is $prop(K)$? Will it converge as $K\rightarrow \infty$? It seems plausible to assume $prop(K)>\frac1n$, but can we say more? My guess would be that $prop(K)$ simply converges to one over the number of lines that survive eventually.

Inspired by a webcomic about role playing games.