I dont know if the New York City code has any specific law which shifts responsibility, but in NY in general you can't force your neighbor to trim the tree over your property or even to do anything on his side. If the neighbor's tree falls and does damage on your property I believe that the neighbor and/or his insurance wont be liable unless you can prove that the tree was already in bad shape (rot, dying, prior damage) which was known to him and which he did not remedy before it caused damage to your property.