"It may be easier for you to put in a claim with your insurance
company. They
in turn will try to get the money from the owner of the tree or the
owner's
insurance. "

No, they won't because unlike you, the insurance company knows that the
property owner of the land that the tree happened to be growing on
isn't responsible for damage to another property the tree fall on,
unless negligence can be proven.

"Request that the neighbor or his insurer pays for the damages. If
not,
get a good book to read while waiting, and meet them in small claims
court"

A waste of time and money. Simpler to do an internet search and see
that the other party has no liability.