No, it does NOT. Depending on the era, Italian manufactured frames also had serial numbers under the bottom bracket. While the Shimano dropouts greatly increase the possibility of it being Japanese, it is not certain. Only the serial number or a country of origin will tell you for sure. And while the dropouts may be Shimano, it does not necessarily mean the derailleurs are. The pic is too poor to tell for sure, but that actually looks more like a SunTour derailleur.

A list of the makes and models of the major components should help narrow down the model. The serial number will determine the year, if it is an Italian frame, though this is unlikely. Assuming the components are original, you can also try to determine component age using the information on the components page of the Vintage-Trek website. The frame features indicate it is probably early 1980s and possibly a Limited, as suggest by Luker.