Did the neighbor have prior knowledge of the tree being 'damaged' ("half dead")?? If so, then he is very probably liable for his negligence in maintaining HIS tree (totally on his property).
I would get a couple of written estimates for repairing the fence and send them (certified RRR) with a letter asking that the neighbor contact you within 10 days to arrange for reimbursement of your damages. If he doesn't make payment, contact your local small claims court and file a complaint.
Note: how old was the fence??? If already needing repair, you probably won't get full replacement value of a new one.
Oh, BTW, be sure to get pictures starting NOW, while the tree is still there. They will help you in court, if needed.