= Binomial branching process
{title2=$f(s)=(1-p+ps)^n$}
A <Galton-Watson process> whose offspring law is a <binomial distribution> $\operatorname{Bin}(n,p)$ has mean $np$ and the displayed offspring <probability generating function>. Its <branching extinction probability> is the smallest root of $q=f(q)$ in $[0,1]$. It dominates a <breadth-first exploration of a binomial random graph> by allowing each explored <vertex> up to $n$ independent possible children, including previously exposed or already discovered <vertices> as phantom children.
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