Suppose $f$ is the probability generating function for the Galton-Watson branching process.

What intuition makes the fact that $f(s) = s$ is the probability of extinction obvious? Moreover, can one find a meaningful interpreation for the "dummy variable", $s$? By meaningful, I mean an interpretation of $s$ which would allow us to write down that the root of $f(s) = s$ yields the extinction probability without needing to expand $f(s)$ to see the result.

Suppose $f$ is the probability generating function for the Galton-Watson branching process.

What intuition makes the fact that $f(s) = s$ is the probability of extinction obvious? Moreover, can one find a meaningful interpretation for the "dummy variable", $s$? By meaningful, I mean an interpretation of $s$ which would allow us to write down that the root of $f(s) = s$ yields the extinction probability without needing to expand $f(s)$ to see the result.

Suppose $f$ is the probability generating function for the Galton-Watson branching process.

What intuition makes the fact that $f(s) = s$ is the probability of extinction obvious? Moreover, can one find a meaningful interpreation for the "dummy variable", $s$?

Suppose $f$ is the probability generating function for the Galton-Watson branching process.

What intuition makes the fact that $f(s) = s$ is the probability of extinction obvious? Moreover, can one find a meaningful interpreation for the "dummy variable", $s$? By meaningful, I mean an interpretation of $s$ which would allow us to write down that the root of $f(s) = s$ yields the extinction probability without needing to expand $f(s)$ to see the result.