Not so fast! I can't think of a good legal theory upon which you could sue and ask a court to order a judgment in your favor. Negligence would require a negligent act or omission, such as failure to trim or remove a weak or diseased tree or limbs. That apparently wasn't the case. Trespass requires an intentional act or omission by the defendant. Limbs and roots that are allowed to grow over the property line are trespasses, as the tree owner is seen as failing to prevent what will certainly occur -- the branches or roots will grow across the boundary.
However, here we have what seems to be an accident - a so-called "act of God" where neither an act or a failure to act on the part of the neighbor can be pointed to as the cause.
Without more facts, I would not say the neighbor has a complete and certain defense, but I likewise don't see an obvious legal theory placing financial responsibility on the neighbor.
