Generally, speaking borderline tree issues are very complicated. You do have rights and should not have to suffer damages as a result of your neighbors tree overhanging your yard or damaging a fence that you own. The tree company is simply acting in self-protect mode and you are not likely to be successful in attempting to force them to cut the tree without the owners permission, they don't want to be in the middle of the battle. You may be able to find a different tree company or you may be able to obtain permission to cut the overhanging limbs from your neighbor. It your neighbor is unwilling to work with you an attorneys demand letter may help to get the process moving before spending time and money on a lawsuit. I recommend you contact our attorneys to discuss your unique facts as well as your rights, remedies, risks, and options before proceeding. You can schedule a free case assessment 24/7 by calling 303-688-0944 or online at https://calendly.com/case-assessment/case-assessment. At your appointment, we'll meet by phone or video (Zoom), we'll discuss your concerns, desired results, rights, remedies, the law, the process, timeline, how we can help, costs, etc.