 Extinction probability of a branching process

If $μ < 1$ , extinction has probability $1$ . For example $p_{0} = 3/4$ , $p_{2} = 1/4$ gives $μ = 1/2$ and $F (s) = 3/4 + s^{2} /4$ ; its only fixed point in $[0, 1]$ is $1$ .
If $μ = 1$ and $p_{1} \neq = 1$ , extinction also has probability $1$ . For example $p_{0} = p_{2} = 1/2$ gives $F (s) - s = (s - 1)^{2} /2$ and $q = 1$ , despite expected generation size in a Galton-Watson process remaining one.
If $1 < μ < \infty$ , extinction has probability $q < 1$ . The value can be zero: if $p_{0} = 0$ , every individual has at least one child, so an initially nonempty family never becomes empty. In particular $p_{2} = 1$ gives $Z_{n} = 2^{n}$ and $q = 0$ .

The critical exception is

p_{1} = 1

: every individual has exactly one child,

F (s) = s

, and

Z_{n} = 1

forever from one ancestor. Thus

μ = 1

but

q = 0

; a blanket claim that every critical Galton-Watson process becomes extinct would be false. These cases concern the Galton-Watson process with its common offspring distribution and independence assumptions. ^[ref]

This probability answers whether any lineage persists, which the offspring mean alone cannot quantify in the supercritical case. A branching process conditioned on extinction uses the same fixed point to reweight reproduction and describe families known eventually to die out.

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