  We present two iterative methods for computing the global and partial
extinction probability vectors for Galton-Watson processes with countably
infinitely many types. The probabilistic interpretation of these methods
involves truncated Galton-Watson processes with finite sets of types and
modified progeny generating functions. In addition, we discuss the connection
of the convergence norm of the mean progeny matrix with extinction criteria.
Finally, we give a sufficient condition for a population to become extinct
almost surely even though its population size explodes on the average, which is
impossible in a branching process with finitely many types. We conclude with
some numerical illustrations for our algorithmic methods.
