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The offspring probabilities $P(K=j)=2^{-j-1}$ for $j\ge 0$ have expected value one. For one initial individual, the branching-process generating-function iteration givesFor $n\ge 1$ 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.

For a Galton-Watson process, condition on the number $K$ of children of the initial individual. The descendants of these children over the next $n$ generations are independent copies of $X_n$. The probability generating function of their total, conditional on $K=k$, is $G_n(s{)}^k$. Averaging over the family-size law givesThis is the branching-process generating-function iteration. Conditioning instead on the last generation gives $G_n(G(s))$; both orders agree because $G_n$ is the $n$th iterate of the same function $G$. The argument holds on the usual PGF domain $|s|\le 1$, with larger domains available when the corresponding expectations converge.