From your question, I cannot tell how long ago you purchased your home. If it's been under a year and half, the roots grew over night and the previous owner did not have the same problem you are now having. Did the previous sellers disclose such a fact in their many disclosures they owed you? (unless you bought at a foreclosure or from a trustee). What does your neighbor say, did he have the same problem with the previous owner? If yes, you may have a claim for misrepresentation or lack of disclosure against the seller, his agent, and possibly your agent if you had one.
The first thing you need to do is hire a professional expert to do a sewer line inspection with a video camera.
In any case, you should try to work it out with your neighbor after you have the inspection results. If the damage stems from his trees, he should abate the situation.
Disclaimer: Ms. Kermani is an attorney licensed to practice in California. This post does not constitute legal advice, should not be relied upon as such, and does not create an attorney client relationship. Since each solution is fact specific, please consult an attorney and provide them with all the facts and documents at issue. Only then can you receive legal advice tailored to your specific situation.
It sounds like this has been an ongoing frustration for you. Why not hire a tree expert to see where the problem is coming from? If it is coming from your neighbor, the expert's report seems to be good evidence to show them and see if you guys can come to an amicable resolution. You are neighbors, after all.
If you do end up taking legal action, at least you will already have the expert's report. It is the plaintiff's burden to prove their case anyhow.
As to whether it is on the City's side, your expert should be able to tell where the damage is coming from although it is not always clear whose responsibility tree maintenance is. As a practical matter, involve both parties in any negotiations or lawsuits.