Why only one new tire? They are replaced in pairs so they are matched. If only one was worn much worse than the other one, a suspension part may be worn that allowed that wheel to sit out-of-alignment. That can also lead to shaking.
Failure to use a click-type torque wrench when installing wheels can result in a warped brake rotor after a few heat-up / cool-down cycles due to uneven clamping forces. Brake rotors can also warp on their own. Your mechanic can measure them to see if that is the cause of the problem. If they are warped, they can be machined unless that will result in them being below the published legal minimum thickness. In that case they must be replaced.