  We consider the images of the initial algebra semantics of weighted tree
automata over strong bimonoids (hence also over semirings). These images are
subsets of the carrier set of the underlying strong bimonoid. We consider
locally finite, weakly locally finite, and bi-locally finite strong bimonoids.
We show that there exists a strong bimonoid which is weakly locally finite and
not locally finite. We also show that if the ranked alphabet contains a binary
symbol, then for any finitely generated strong bimonoid, weighted tree automata
can generate, via their initial algebra semantics, all elements of the strong
bimonoid. As a consequence of these results, for weakly locally finite strong
bimonoids which are not locally finite, weighted tree automata can generate
infinite images provided that the input ranked alphabet contains at least one
binary symbol. This is in sharp contrast to the setting of weighted string
automata, where each such image is known to be finite. As a further
consequence, for any finitely generated semiring, there exists a weighted tree
automaton which generates, via its run semantics, all elements of the semiring.
