  A curious connection exists between the theory of optimal stopping for
independent random variables, and branching processes. In particular, for the
branching process $Z_n$ with offspring distribution $Y$, there exists a random
variable $X$ such that the probability $P(Z_n=0)$ of extinction of the $n$th
generation in the branching process equals the value obtained by optimally
stopping the sequence $X_1,...,X_n$, where these variables are i.i.d
distributed as $X$. Generalizations to the inhomogeneous and infinite horizon
cases are also considered. This correspondence furnishes a simple `stopping
rule' method for computing various characteristics of branching processes,
including rates of convergence of the $n^{th}$ generation's extinction
probability to the eventual extinction probability, for the supercritical,
critical and subcritical Galton-Watson process. Examples, bounds, further
generalizations and a connection to classical prophet inequalities are
presented. Throughout, the aim is to show how this unexpected connection can be
used to translate methods from one area of applied probability to another,
rather than to provide the most general results.
