This is one of those things that get filed away under, "Didn't know until I had to." That said, it's always best to know liability and fault before something occurs, especially if your property contains big trees. If you rent, you're off the responsibility hook as your landlord will take care of things for you.
Do you know whose responsibility it is when one of your trees fall? What about when your neighbor's tree pummels your new fence and swing set? Here's a quick, general guide just to put you in the know:
YOUR TREE + YOUR HOUSE: your insurance pays to remove the tree and for the damage to your home.
YOUR TREE + NEIGHBOR'S HOUSE: neighbor's insurance pays.
NEIGHBOR'S TREE + YOUR HOUSE: your insurance pays.
YOUR TREE + YOUR YARD + DOESN'T HIT ANYTHING: you pay out of pocket for the removal.
You can check out more information on situations like this over at CBS Philly (via The Consumerist). It always better to know whose insurance should be paying what, especially if you fear your neighbors might not have any!