= Solution
For $|s|\leq1$, all the generating-function expectations are absolutely bounded. Conditional on $X=k$, independence gives
$$
\mathbb E[s^T\mid X=k]=s\prod_{i=1}^k\mathbb E[s^{T_i}]=sG_T(s)^k.
$$
At $k=0$ the product is one, consistent with the empty-family case. The <law of total expectation> now yields
$$
\boxed{G_T(s)=s\sum_{k=0}^{\infty}p_kG_T(s)^k
=sG_X(G_T(s))=s\exp\{\lambda(G_T(s)-1)\}.}
$$
This is the <total progeny generating function of a branching process>. The given almost-sure finiteness of $T$ ensures $G_T(1)=1$. The conditional-expectation proof avoids assuming a finite expected total before it has been established.
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