These maps are not accurate. You need a survey. After the survey is done (maybe neighbor will share the cost), perhaps you can execute a boundary line agreement and agree to the lines shown on the survey. This will prevent future claims and litigation. A local attorney can help you with the agreement and it shouldn't be expensive.
Actively practicing law in Texas. Inactive licenses in Arizona and Georgia. All answers are general in nature and no attorney/client relationship exists in this forum.
We have focused on land disputes for almost three decades and this is the first time a neighbor has claimed to give land back! First, understand that ancient fence lines provide suggestions of boundaries, but are not true demarcations. They are often the basis for long term claims for adverse possession, but your neighbor is admitting your ownership, here. Your picture was very informative, but the next step is to obtain a survey based on deed language, record plats, and any stakes or other monuments the surveyor can locate. Your rights will flow from the results of that survey. Now, back to the tree-- Your neighbor's concern becomes your problem if the tree is on your land, and it actually falls and does damage. Until then, it is just the impetus for you to examine the underlying title issue. Did you purchase title insurance? Did you obtain a survey when you purchased?
Practically speaking, you might spend far too much time and money fighting over the land than what it would cost to be a good neighbor and cut the tree down. It certainly seems like you're not making a claim to this extra land, so you might really want to think about solving the tree problem instead of litigating this issue. In the end, you'll spend less money and have a better relationship with your neighbor as well.
DISCLAIMER: Brandy A. Peeples is licensed to practice law in the State of Maryland. This answer is being provided for informational purposes only and the laws of your jurisdiction may differ. This answer based on general legal principles and is not intended for the purpose of providing specific legal advice or opinions. Under no circumstances does this answer constitute the establishment of an attorney-client relationship. For legal advice relating to your specific situation, I strongly urge you to consult with an attorney in your area. NO COMMUNICATIONS WITH ME ARE TO BE CONSTRUED AS ARISING FROM AN ATTORNEY-CLIENT RELATIONSHIP AND NO ATTORNEY-CLIENT RELATIONSHIP WILL BE ESTABLISHED WITH ME UNLESS I HAVE EXPRESSLY AGREED TO UNDERTAKE YOUR REPRESENTATION, WHICH INCLUDES THE EXECUTION OF A WRITTEN AGREEMENT OF RETAINER.
If the tree is on your property, you are responsible for maintaining the property and for ensuring that the tree is maintained. The real question here is whether the area outside of both your and your neighbor's fence is your property. The only way to establish the true property lines is to have a boundary survey done. It will cost a lot, but it is necessary if you cannot otherwise establish boundaries.