Based on observations X1, . . . , Xn of successive generations of a discreteparameter Galton-Watson branching process, one wishes to predict whether extinction or explosion will ultimately occur. This problem can be formulated as a simple hypothesis-testing problem to which the Neyman-Pearson Lemma is directly applicable if the extinction probability is known or estimable. If it is not, valid (but conservative) tests still can be obtained. The cases where n is fixed or determined sequentially are both considered.