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We consider a class of multitype Galton-Watson branching processes with a countably infinite type set X_d whose mean progeny matrices have a block lower Hessenberg form. For these processes, the probability q(A) of extinction in subsets of types A⊆X_d may differ from the global extinction probability q and the partial extinction probability q̃. After deriving partial and global extinction criteria, we develop conditions for q<q(A)<q̃. We then present an iterative method to compute the vectorarXiv:1805.07634v2 fatcat:pwzf77u5tbhbpjhdpc5vq2yfji