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probability, varying environment, inhomogeneous Galton-Watson process, prophet inequal-ities. 1 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 ospring distribution Y, there exists a random variable X such that the probability P (Z n = 0) of extinction of the nth generation in the branching process equals the value obtained by optimally stopping the sequence