  The HOM-problem, which asks whether the image of a regular tree language
under a tree homomorphism is again regular, is known to be decidable by [Godoy,
Gim\'enez, Ramos, \`Alvarez: The HOM problem is decidable. STOC (2010)].
Research on the weighted version of this problem, however, is still in its
infancy since it requires customized investigations. In this paper we address
the weighted HOM-problem and strive to keep the underlying semiring as general
as possible. In return, we restrict the input: We require the tree homomorphism
h to be tetris-free, a condition weaker than injectivity, and for the given
weighted tree automaton, we propose an ambiguity notion with respect to h.
These assumptions suffice to ensure decidability of the thus restricted
HOM-problem for all zero-sum free semirings by allowing us to reduce it to the
(decidable) unweighted case.
