A weighted tree transformation is a function : T T ! A where T and T are the sets of trees over the ranked alphabets and , respectively, and A is the domain of a semiring. The input and output product of with tree series ' : T ! A and : T ! A are the weighted tree transformations ' / and . , respectively, which are de ned by (' / )(t; u) = '(t) (t; u) and ( . )(t; u) = (t; u) (u) for every t 2 ...