If the branch that falls on the neighbor’s property belonged to a healthy tree and the tree owner had no reason to think the branch might come down, then generally the tree owner is not financially responsible for the damage caused by the falling branch. This situation usually arises in the context of a storm with strong winds or a tornado. A sufficiently bad storm can break off a healthy tree branch.
The law’s reasoning here is that the owner wasn’t careless and didn’t do anything wrong. It was just bad luck that the branch fell. The tree owner wasn’t able to foresee that it would come down in the storm.