Tree Hugger Dilemma

My clients are hardworking immigrants. They bought new construction in an established neighbohrood. Now they want to build a garage behind their house. They have the permits. Their contractor is ready to go. However, he warned them that a big old tree--80% on their property, 20% on the neighbor's property--will be destabilized when they start construction. It has to come down. The neighbor keeps ignoring their attempts to talk. She is a tree hugger. Local government will not get involved. They say it is a dispute between neighbors. My clients have offered to plant new trees and to put up privacy fencing. The tree hugger neighbors have 3 big, noisy dogs and my clients never complain about them. They are quite anxious to start construction. They just asked for my advice. They also asked for the name of an attorney. I really don't know what to tell them to do. Except I cautioned them not to make the neighbor mad. Any suggestions????