We had an arborist tell us that an ancient elm tree which was struck by lightning was dead and needed to be cut down. Our insurer had their own arborist come out and told us the tree was okay.
The tree was okay - and is still standing many years after the reports from the dueling arborists, a perfectly healthy (and beautiful) tree.
I would not act on a single arborist report, especially if the neighbor's arborist also offers tree removal services.
I might wait to see if your former neighbor is intent on pursuing the tree's removal or if the neighbor's original complaint about the tree was less about the tree and more about his dislike of you.
Ironically this neighbor had a dispute with the previous owner as well that she didn't disclose to me when I bought myself. She actually told me after I brought up the tree issue (the previous owner and I stayed in touch after the sale) that she had wanted the tree cut down but he told her he DID NOT want it cut down as his kids play on it. I have a recorded phone call of this statement..
Your point is very valid, yes, the arborist also did tree removal services, had been in business for 6 months and was hired from 120 miles away when dozens of other arborists with well established businesses were around the corner. That was my entire point about the guys credentials and distance, it seemed suspect to me, like shopping for an arborist trying to establish his business and getting him to just say whatever you want.
Also note, the arborist illegally came onto my property to assess the tree. I was not aware of this and did not give permission for him to do this. Under the fruit of the poisonous tree doctrine, I would think that report would be inadmissible in court anyways, no?
Heck, the property has other trees on it that are leaning (trees don't always grow exactly straight up) that this neighbor didn't send me a notice about. Ultimately the new owner is responsible for the trees, no? Are you of the opininon that I should let this sit or should I send the letter to cover my butt?
One concern I have about sending a letter through an attorney or certified mail is that they would then have a way to track me down address or attorney wise. They have my phone number but not my new address.