If you notified your neighbor via registered letter (or have some other means that documents their receipt of the 'notice' you sent like a text or email) that the tree was unhealthy, you may be able to have your insurance company subrogate (go after their insurance company). It will be especially helpful if you have photos of the tree before and after the damage was incurred. If the neighbor claims to have no idea the tree was failing, they will likely claim it was an 'act of God" and have no liability for damages to your property. Get as many close-up photos of the tree possible showing the rot and all the damage it caused.