 Total progeny generating function of a branching process

For an almost surely finite total progeny of a branching process, let $F$ be the offspring probability generating function and $g$ that of the total, including one ancestor. Conditional on $k$ children, the total is one plus the sum of $k$ independent descendant totals. Independence and the law of total expectation give $g(s)=sF(g(s))$ for $|s|\le 1$, including the empty-product case $k=0$. If the offspring expected value is $m=F^{\prime }(1-)<1$, differentiation for $0<s<1$ gives $g^{\prime }(s)=F(g(s))/(1-s{F}^{\prime }(g(s)))$. The denominator is positive because $F^{\prime }(g(s))\le m$. Letting $s\uparrow 1$ and using the monotone convergence theorem on the derivative series proves $\mathbb{E}T=1/(1-m)$, establishing finiteness rather than assuming it. For Poisson branching process offspring, $F(z)=e^{m(z-1)}$.