Assuming the tree's truck is on your property, it's yours, and generally you're responsible for its maintenance, and for any damage it may cause to your neighbors. If the neighbor wants to trim branches that overhang their property, they're free to do that, but they have to be careful not to harm the tree.
In the interests of good neighbor relations, since you do have to live next to this person, you may want to offer to pay for the tree trimming, and take their preferences into account, but exercise control over the work to make sure your tree isn't harmed.
PLEASE READ THIS BEFORE YOU COMMENT, EMAIL ME OR PHONE ME. I'm only licensed in CA. This answer doesn't make me your lawyer, and neither do follow-up comments and/or emails and/or phone calls, and you shouldn't expect me to respond to your further questions if you haven't hired me. We need an actual agreement confirmed in writing before any attorney-client relationship is formed. This answer doesn't constitute legal advice, and shouldn't be relied on, since each state has different laws, each situation is fact specific, and it is impossible to evaluate a legal problem without a comprehensive consultation and review of all the facts and documents at issue.