Acorns and bird droppings

When my neighbor moved in a year ago and he cut down 3 trees that were on his side of mutual fence. He also trimmed the branches that were overhanging his side of the fence from a tree that is in my yard.

Today he sent me some photos of acorns and bird droppings that he claims are coming from my tree. The same tree he trimmed last year.

I am curious before I speak to him if I have any responsibilities for stuff falling in his yard from a tree that is mostly in my yard. It is my understanding that he can only trim the branches that overhang his side of the fence. Also since I am sure he is determined to keep the bird droppings from falling in his yard is there anything I should do to document the health our my tree now in case he damages it at any time in the future. The tree is one of the favorite parts of my yard and I would hate for it to get damages because my neighbor doesn't really seem to like nature.