= Geometric-offspring critical branching process
{title2=$G(s)=1/(2-s)$}
The offspring <probabilities> $P(K=j)=2^{-j-1}$ for $j\geq0$ have <expected value> one. For one initial individual, the <branching-process generating-function iteration> gives
$$
G_n(s)=\frac{n-(n-1)s}{n+1-ns},\quad P(X_n=0)=\frac n{n+1},\quad
P(X_n=j)=\frac1{(n+1)^2}\left(\frac n{n+1}\right)^{j-1}\ (j\geq1).
$$
For $n\geq1$ the <PGF> disk is $|s|<1+1/n$; $G_0(s)=s$ separately. Conditional on survival, the population is positive geometric with success <probability> $1/(n+1)$, leading to the <exponential limit of surviving geometric branching>.
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