I think in most states the responsibility depends on the condition of the tree and whether or not the original owner is negligent.
When a healthy tree blows down on another person's property it is considered and "act of god" and the portion of the tree on your property becomes your property and responsibility. Seems unfair but that is the law as I found out when my neighbor's trees fell on my fence.
If the tree is unhealthy and the property owner was notified or aware of it before it falls then they are negligent in not preventing the damage.