Does the tire tread reduce the likelyhood of getting a puncture?
It depends. Tread pattern affects puncture probability in two opposite ways:


Slick tires often, but not always, have thinner rubber, which offers less protection.
Knobs are generally too thick to be punctured through and put some distance between ground and the flat parts of the tire, but on the other hand the part between knobs may be very thin.
Tires with grooved tread may be punctured because small pieces of gravel or glass get stuck in grooves and eventually dig through the rubber. This happens very rarely with slick or knobby tires.


In your case, you should be worried about pinch flats, which happen when the tire is pinched between a rock and rim. 28mm may be just enough for smooth cobblestone roads but is very narrow for rocky or soft terrain.