Mean and variance of a Galton-Watson generation ($\mathbb{E}[X_n]={\mu }^n,v_{n+1}={\mu }^2{v}_n+{\sigma }^2{\mu }^n$)

With one initial ancestor and independent offspring counts of finite mean and variance, conditioning on the current generation gives the displayed recurrences. For $\mu \ne 1$ the variance is ${\sigma }^2{\mu }^{n-1}({\mu }^n-1)/(\mu -1)$ for positive generation index. At $\mu =1$ it is $n{\sigma }^2$. At zero offspring mean the distribution is concentrated at zero and later generations vanish. These formulas are for the Galton-Watson process, not arbitrary dependent reproduction.