Extinction probability of a branching process ($q=\mathbb{P}(\exists j:Z_j=0)$)

For a Galton-Watson process with one ancestor this is the probability that some generation is empty. It is the smallest fixed point of the offspring probability generating function in $[0,1]$; the complementary survival probability is $1-q$. A branching process conditioned on extinction reweights reproduction using this fixed point.

For a Galton-Watson process started from one ancestor, this is the probability that every generation is nonempty. It is the complement of the branching extinction probability. The binomial branching survival correction distinguishes finite-binomial reproduction from the Poisson branching process near mean one.