  We consider weighted tree automata over strong bimonoids (for short: wta). A
wta $\mathcal{A}$ has the finite-image property if its recognized weighted tree
language $[\![\mathcal{A}]\!]$ has finite image; moreover, $\mathcal{A}$ has
the preimage property if the preimage under $[\![\mathcal{A}]\!]$ of each
element of the underlying strong bimonoid is a recognizable tree language. For
each wta $\mathcal{A}$ over a past-finite monotonic strong bimonoid we prove
the following results. In terms of $\mathcal{A}$'s structural properties, we
characterize whether it has the finite-image property. We characterize those
past-finite monotonic strong bimonoids such that for each wta $\mathcal{A}$ it
is decidable whether $\mathcal{A}$ has the finite-image property. In
particular, the finite-image property is decidable for wta over past-finite
monotonic semirings. Moreover, we prove that $\mathcal{A}$ has the preimage
property. All our results also hold for weighted string automata.
