Sheesh! Why is that guy (Smitty's neighbor) freaking out over some branches? Some people...
I've got exactly the opposite situation: the offending trees are on my property, and the branches overhang my neighbor's lawn. She's an elderly widow. When the branches cause trouble, she tells me and I trim them. No big deal.
Trees have strange rules, as others have pointed out. Here's a couple of examples from NJ:
A neighbors tree fell during a storm, crossing the property line. No damage was done. After contacting insurance companies, the parties learned that they were each responsible for the part of the tree that was lying on their respective properties. However, since no damage was done, no insurance claim would be paid.
During another storm, my one of my neighbors trees fell across my property line and smooshed my fence. My insurance company paid to clean up the part of the tree on my side of line (they left me all of the stuff good for firewood), and repaired my fence. My neighbors insurance company didn't pay anything, because there was no damage on his side. I did help him cut tree into firewood.