Neighbor's Trees
Learn how to discuss tree issues directly with your neighbor. The Pierce County Center for Dispute Resolution can:
Provide advice and tips on how to best approach the neighbor
Sit down with you and your neighbor to help you discuss and resolve the issue
Tacoma-Pierce County Bar Association may be able to assist with legal questions such as what your rights are as a neighbor in regard to trees.

Please note that trees on private property or neighboring rights-of-way that do not pose a significant threat to public safety, health or welfare are a private matter to be handled by the involved property owners. The City cannot get involved in private disputes. Please direct your questions elsewhere, such as to those listed above, for these situations.

Quoted:
Find the property line and use a ditch witch to dig a trench as deep as needed to sever the roots on your side. Problem solved as far as you're concerned, his dying tree is now his issue.

$200 a day for a 30" deep ditch witch with just a couple mins search for local equipment rentals. Might find a better deal or a deeper trencher if you  call around.

You'll wanna check for utilities and pipes beforehand.