Context-free tree grammars have rules that allow to replace a non-terminal node in a tree by a whole tree. They are widely used in different areas in computer science. (Non-strict) tree adjoining grammars, originating from the examination of natural languages, only allow the restricted replacement of a node in a tree by a complete tree drawn from a finite collection. Their importance in linguistics stems from the fact that they are strictly more expressive than context-free (string) languages and seemingly sufficient to describe natural languages, but still efficiently parsable. The two grammar types are known to be weakly equivalent in the sense that they describe the same class of string languages. The present paper shows that they are also strongly equivalent: they generate the same class of tree languages.