A tree limb fell off a neighbors tree and hit our fence and roof, putting a hole in it. They had to pay to fix it (them or their insurance, don't remember which did), because the tree was their property since it was on their property and it should have been trimmed. Apartment property might be different, but I still believe it should be the property owners job to take care of lawn maintenance, because you don't own the lawn they do. Every apt I have lived in, the landlord sends yard people around once a month to do the grass and trim bushes and stuff.

Just saying this is my personnel experience with it. I would check with someone other than the property owner first just to make sure he don't try to screw you over or tell you its not his fault.