Total progeny of a branching process ($T={\sum }_{j\ge 0}{Z}_j$)

The total progeny counts all individuals, including the initial ancestor, in a Galton-Watson process. It is infinite on survival. If the offspring mean is $m<1$, the expected generation size in a Galton-Watson process and the monotone convergence theorem give $\mathbb{E}T=1/(1-m)$.