If I were you I would argue this as maintenance of a jointly owned wall, with each neighbor responsible for 50% of the cost. If I were your neighbor I would argue that it was encroachment by your tree roots which caused the damage, and that you are thus responsible for the full cost (your 50%, plus the damage to my 50% caused by the encroaching tree roots).

I would personally start by making sure that I knew where the property line lies, so as to be sure that this is a shared wall. I would not agree to use a contractor chosen by the neighbor, but would insist upon a mutually acceptable contractor.
Quote Quoting California Civil Code § 841.
Coterminous owners are mutually bound equally to maintain:

1. The boundaries and monuments between them;

2. The fences between them, unless one of them chooses to let his land lie without fencing; in which case, if he afterwards incloses it, he must refund to the other a just proportion of the value, at that time, of any division fence made by the latter.