# Deeper intuition about probability generating function through extinction probability?

For a Galton-Watson branching process, if $$f$$ is the p.g.f of the process, then the extinction probability, $$q$$, is given by the root to $$f(q) = q$$.

This can be seen intuitively simply by writing out, if $$X$$ is the progeny of the root ancestor:

$$q = \sum_{i = 0}^{\infty} q^i P(X = i)$$

This gives the probability that each of the $$i$$ descendants goes extinct scaled by the probability of having $$i$$ descendants.

Is there a deeper intuition to the p.g.f for understanding why the extinction probability is given so elegantly? Is there more going on with $$f(s), s \in [0,1]$$?