About insurance:
There are two types of liability insurance for businesses or professionals: general liability insurance and "E&O" (errors and omissions) insurance. If your arborist caused damage to you or your property by botching the job, that falls under E&O insurance, which the arborist might or might not have. If instead you tripped over his equipment and broke your ankle, that would come under general liability.
Ultimately you can't claim against his insurance, you have to send him your complaint and request for compensation, and ultimately sue him if you can't otherwise get satisfaction. His insurance isn't really your business, though it's nice to know if there is enough money available (either his own assets or insurance) to pay if you win in court.
Since you are not qualified to identify the cause of your tree's problems, you won't be able to prove anything unless you get a consultation from a good forensic arborist, so that's the obvious first step in figuring out both who/what is to blame and how/whether the tree can be saved.
If the tree was ailing before your arborist arrived, and he was an actual arborist rather than just a tree surgeon, he should have told you about your tree's problems before agreeing to do work on the tree. Or at least mentioned it while he was doing the work, if that was when he noticed it. Did he mention anything? If not, Raintree might be onto something. There are sneaky ways to kill a tree if someone wants to. Copper nail(s) driven into the trunk near the base, for one.
