In this paper we investigate the relationship between classes of tree-to-tree-series (for short: t-ts) and o-tree-to-tree-series (for short: o-t-ts) transformations computed by restricted deterministic bottom-up weighted tree transducers (for short: deterministic bu-w-tt). Essentially, deterministic bu-w-tt are deterministic bottom-up tree series transducers [EFV02, FV03, ful, FGV04], but the former are de ned over monoids whereas the latter are de ned over semirings and only use the multiplicative monoid thereof. In particular, the common restrictions of non-deletion, linearity, totality, and homomorphism [Eng75] can equivalently be de ned for deterministic bu-w-tt. Using well-known results of classical tree transducer theory (cf., e.g., [Eng75, Fül91]) and also new results on deterministic bu-w-tt, we order classes of t-ts and o-t-ts transformations computed by restricted deterministic bu-w-tt by set inclusion. More precisely, for every commutative monoid we completely specify the inclusion relation of the classes of t-ts and o-t-ts transformations for all sensible combinations of restrictions by means of inclusion diagrams.

In this paper we investigate the relationship between classes of tree-to-tree-series (for short: t-ts) and o-tree-to-tree-series (for short: o-t-ts) transformations computed by restricted deterministic bottom-up weighted tree transducers (for short: deterministic bu-w-tt). Essentially, deterministic bu-w-tt are deterministic bottom-up tree series transducers [EFV02, FV03, ful, FGV04], but the former are de ned over monoids whereas the latter are de ned over semirings and only use the multiplicative monoid thereof. In particular, the common restrictions of non-deletion, linearity, totality, and homomorphism [Eng75] can equivalently be de ned for deterministic bu-w-tt. Using well-known results of classical tree transducer theory (cf., e.g., [Eng75, Fül91]) and also new results on deterministic bu-w-tt, we order classes of t-ts and o-t-ts transformations computed by restricted deterministic bu-w-tt by set inclusion. More precisely, for every commutative monoid we completely specify the inclusion relation of the classes of t-ts and o-t-ts transformations for all sensible combinations of restrictions by means of inclusion diagrams.

Tree series transformations computed by bottom-up and top-down tree series transducers are called bottom-up and top-down tree series transformations, respectively. (Functional) compositions of such transformations are investigated. It turns out that the class of bottomup tree series transformations over a commutative and complete semiring is closed under left-composition with linear bottom-up tree series transformations and right-composition with boolean deterministic bottom-up tree series transformations. Moreover, it is shown that the class of top-down tree series transformations over a commutative and complete semiring is closed under right-composition with linear, nondeleting top-down tree series transformations. Finally, the composition of a boolean, deterministic, total top-down tree series transformation with a linear top-down tree series transformation is shown to be a top-down tree series transformation.

Tree series transformations computed by bottom-up and top-down tree series transducers are called bottom-up and top-down tree series transformations, respectively. (Functional) compositions of such transformations are investigated. It turns out that the class of bottomup tree series transformations over a commutative and complete semiring is closed under left-composition with linear bottom-up tree series transformations and right-composition with boolean deterministic bottom-up tree series transformations. Moreover, it is shown that the class of top-down tree series transformations over a commutative and complete semiring is closed under right-composition with linear, nondeleting top-down tree series transformations. Finally, the composition of a boolean, deterministic, total top-down tree series transformation with a linear top-down tree series transformation is shown to be a top-down tree series transformation.