Weighted Bottom-Up and Top-Down Tree Transformations Are Incomparable.

Weighted bottom-up and top-down tree transformations over commutative semirings are investigated. It is demonstrated that if the range of a weighted bottom-up tree transformation is well-defined (i.e., for every output tree there are only finitely many input trees that can transform to the output tree with nonzero weight, so that all involved sums remain finite), then the range is hom-regular, which means that it is the image of a regular weighted tree language under a tree homomorphism. Additionally, the strictness of the first level of the weighted bottom-up and top-down tree transformation hierarchy is proved, which was open for any ring.