Expected generation size in a Galton-Watson process ($\mathbb{E}[Z_n]=m^n$)

If a Galton-Watson process starts from one individual and has offspring mean $m$, conditioning on generation $n$ gives $\mathbb{E}[Z_{n+1}\mid Z_n]=m{Z}_n$. Induction therefore gives $\mathbb{E}[Z_n]=m^n$.