I do not think you have a recourse if a neighbour starts cutting branches of your tree overhanging his property. He has a right to enjoy his property and you have no right to obstruct him from doing so.

It would be prudent for the neighbour from a legal point of view to give you a notice to remove the over-hanging branches at your expense within a period of, say, 3 weeks, failing which he would get it done on his own and pay the expenses and send you the bill in this regard to recover the expenses from you which you would be under a legal obligation to pay and for non-payment of which he can file a suit of recovery against you.

Your neighbour has full rights to the space above his land. You have no right to compromise his rights in the garb of your supposed "right to protect the life of your tree".

--M C Gupta

agreed rightly advised

You may seek permission from the appropriate authorities for cutting such portion of the tree as are branching out of your property.