Quasi-alphabetic tree bimorphisms [Steinby, Tîrnă ucă: Defining syntax-directed translations by tree bimorphisms. Theor. Comput. Sci., to appear. http://dx.doi.org/10.1016/j.tcs.2009.03.009 , 2009] are reconsidered. It is known that the class of (string) translations defined by such bimorphisms coincides with the class of syntax-directed translations. This result is extended to a smaller class of tree bimorphisms namely (linear and complete) symbol-to-symbol tree bimorphisms. Moreover, it is shown that the class of simple syntax-directed translations coincides with the class of translations defined by alphabetic tree bimorphisms (also known as finite-state relabelings). This proves that alphabetic tree bimorphisms are not sufficiently powerful to model all syntax-directed translations. Finally, it is shown that the class of tree transformations defined by quasi-alphabetic tree bimorphisms is closed under composition. The corresponding result is known in the variable-free case. Overall, the main results of [Steinby, Tîrnă ucă] are strengthened.