Translation:Who hunts for deer among the trees?
I believe it has to do with the generality of the term. "Birds" or "madarak" is a whole class of animals, with many many species included, so it somehow implies multiple possible targets. On the other hand, "szarvas" is a specific kind of mammal, not the whole class of mammals. Especially in Hungary (or Europe), it usually means one (or maybe a very few) specific deer species.
So, to put it in another context, it is like "fruits and bread". Why is fruits in the plural and bread in the singular?
If we were hunting for pheasants ("fácán"), it would most probably be used in the singular, even though usually it involves a mass killing.