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?