= Solution
For $0\le s\le1$, condition on $N=k$. Independence of the descendant <total progeny> variables gives
$$
\mathbb E\!\left[s^{\sum_{i=1}^N Z_i}\mid N=k\right]=\prod_{i=1}^k\mathbb E[s^{Z_i}]=G_Z(s)^k,
$$
including the empty product $1$ when $k=0$. Taking <expectation> over $N$ proves the <probability generating function of an independent random sum> formula
$$
\boxed{\mathbb E\!\left[s^{\sum_{i=1}^N Z_i}\right]=\sum_{k=0}^\infty\mathbb P(N=k)G_Z(s)^k=G_N(G_Z(s)).}
$$
The same identity holds for complex $|s|<1$ by absolute convergence.
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