For supercritical finite mean Galton-Watson as well as for age-dependent
 branching processes
it is known that there are always some norming constants making the
normed process converge almost surely. Recently, Biggins and Kyprianou [3] have proven
that for a supercritical branching random walk, convergence in probability
to a nondegenerate variable obtains for suitable norming.
We show that the tail of the limiting distribution as well as the growth of the Galton-Watson process of ge
neration sizes may
determine the form of the norming constants and provide a criterion for a.s. convergence.