If I am understanding your question, you are asking who owns the tree if most of the tree is on your property. The tree belongs to whoever has the trunk originating on their property, since that is how it is determined where the tree lies in regard to property lines. When in doubt discuss with your neighbor and/or a property evaluator of some sort. When discussing YOU doing any type of pruning, cutting, trimming, treatment to the tree, get everything in writing from the owner of the tree (as in, if they say it's ok to do anything to the tree, they have to write it and sign it as proof that they gave permission.) And even then, you may (and probably will) be liable for any damages that occur to the tree and their property if something goes awry. It's a very touchy subject. I would imagine that a good option if you want anything done to the tree that the neighbor agrees to but doesn't want to pay for, is to have the owner hire a licensed arborist, and you pay for the work. That way if anything happens, it's the arborist and the owners responsibility legally. This is all my best guess though. Hope it helps