WAV, a piece can be both composed and traditional. Provided it has been adopted into the oral tradition and adapted by it, it becomes traditional. The fact that we may know the identity of the original composer doesn't alter that. However where a piece is adopted without being changed, then it is not traditional.

All music was composed by someone, it is pure chance whether or not we know their identity. Some pieces have been passed on through the oral tradition for hundreds of years. If some diligent researcher now turns up the original composer, does that make them any less traditional? I don't think so.