Can the 700 x 38c tire on the Trek DS2 be used on the Trek Verve 3 which comes with a 700 x 45?
The most critical dimension is the bead-seat diameter of the tires, which is the same (denominated here as "700", which really means 622 mm).
The second dimension is the width. Any given rim can accept a range of tire widths. Looking at Trek's website, it's not clear exactly what rims are on either of these bikes, so it's hard to know what tires they'll accommodate. If the rim on the Verve is carrying a tire at the narrow end of its range, and the rim on the DS2 is carrying a tire at the wide end of its range, then they might not be interchangeable. But I recommend trying it out.
The third dimension to consider is tire clearance with respect to the frame and brakes. Since the DS2 has disk brakes and the Verve has cantilever brakes, clearance shouldn't be a problem.
Went to my local Trek store yesterday, nice folks there to be sure. I explored the question with them, they said "sure, it will work fine", tried the 700 x 38c on the Verve, it was a great ride, compared to the 700 x 45. The 700 x 38 ride was smoother, faster and easier maneuvering. Over tough terrain I suspect the 700 x 45 would be a better tire. Thanks for your response.