There are orchards, there are even some plantations, but this is about trees around the house.
It is common to have a bit of land around the house, specially in the suburbs, and some trees on it. The lawn may be manicured, but just as often it is used to grow flowers. The trees on it, however, may be any indigenous sort (acacia, chestnut, lombardy poplar, maple, linden, beech, birch), or decorative (fir, pine, thuja).
In the backyard, however, it would make no sense, in the mind of a Serbian host, to grow decorative trees, when the space can be better used to grow fruit. It can be anything - cherries, cherries (there are separate words for sweet and sour cherries, as they are separate species), apricots, walnuts, apples, pears, quinces, plums, peaches, almonds. Often one can find a shade made out of grapevine, so in the fall you sit at a table outside and eat grapes straight off the vine.
Mulberry was particularly popular as a tree on the street, or to line up the streets and roads (specially good against snow covering the roads), but it was largely eradicated after a politician died hitting one in the early sixties, and also in an attempt to get rid of the mulberry worm - which then spread to other fruit. It's back now, but not in such numbers.
With so many people producing fruit, moonshine is quite common (and legal, btw).