looked up some law regarding trees and discovered something called "air
rights" or something like that, which basically provided that the branches
of a tree, as long as it doesn't physically touch a neighbors property may
not be cut by that neighbor even if it is "over" their property. I was
wondering if there are any comparable laws regarding the tree trunk itself
and its major roots, which obviously run well under my neighbors property?
Help would be greatly appreciated -- thanks.

Basically, although laws will vary, anything that encroaches on their property,
whether on, under, or above the ground, can be removed by them (but must be
given back to you, or else they may be guilty of the tort of conversion).

Too bad for the tree. Try to talk them into letting it live.