The aim of this paper is to start investigations on the possibilities of introducing a structure in the strings generated by internal contextual grammars. First, we consider the idea of associating a tree to a derivation in such a grammar. This can be done in a natural way, by associating parentheses to the contexts of the grammar. In this way we obtain a restriction on the derivation in a contextual grammar, as well as a direct manner of defining the ambiguity of contextual grammars. Then, we consider a relation on the set of symbols appearing in a string, in the sense already used in descriptive linguistics. By starting from a set of axioms which are structured strings and adjoining to them contexts as usual in contextual grammars, but having prescribed dependences between their symbols, we obtain a set of structured strings. By imposing conditions on the structure of the strings (crossed-noncrossed dependences, a tree structure, a link structure in the sense of link grammars, etc), we obtain a restriction on the derivation in a contextual grammar, as well as a direct manner of defining the structure of languages generated by contextual grammars. The linguistic relevance of these structures associated to strings generated by contextual grammars remains to be further explored.