Rounds has defined the class of context-free tree languages and proved its closure properties under linear homomorphism and intersection with regular sets of trees. The open problem of closure under linear inverse homomorphism has just been solved negatively. As for context-free grammars we can define Greibach's forms for context-free grammars of tree languages. Unfortunately there are context-free tree languages without Greibach's grammar. We define tree grammars without left (or on the top) recursivity and show they generate exactly the class of Greibach's tree languages. Then we prove the closure of that class under linear inverse homomorphism. This result proves that Greibach's tree languages are strictly contained in context-free tree languages.
