I saw on a TV judge show, that you are allowed to prune back any tree branches that hang over your property line. Not only do you have the right, but it is your responsibility to maintain your own property in this way. If you do not prune back your neighbors tree, and the hunging branch does damage to your house, it's considered your own fault. You cannot sue your neighbor for the damage. It is your responsibility to cut back any roots before they damage your foundation.
Too bad if your neighbor doesn't like it. he can't stop you from cutting anything that crosses your property line. If he doesn't like it, maybe you can work a deal to split the cost of having the entire tree completely removed.