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?

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