Offspring distribution of a branching process ($f(s)={\sum }_{j\ge 0}{p}_j{s}^j$)

In a Galton-Watson process, each individual independently has $j$ children with probability $p_j$. This probability distribution determines its offspring probability generating function, its mean $f^{\prime }(1)$ when finite, and the branching extinction probability. A binomial branching process and a Poisson branching process have different offspring distributions even when their means agree.