CA renter is right–in CA one cannot simply cut the roots and the branches at the property line. If it kills the tree, you may be liable.
This is a perfect case of where the two property owners ought to get together and work it out. The tree’s owner needs to be appraised of the damage being done and the liklihood of it increasing in the future. Verbally at first, then in writing if no action taken, with the letter ending “…I hope we can solve this amicably and I do not have to impose on you the additional expense of my attorney.”