You will be solicited by tree services after the storm. The single easiest way to determine if the person is qualified to do tree work on your property is to ask them what industry standards for working and safety they are bound by. If they can't answer without hesitation, move on to the next.
Customer: "What industry standards for work and safety are you bound by?"
Solicitor: "ANSI A300 and Z133!"
Even if they just answer "ANSI" you are ahead of the curve. You can even just ask them what ANSI stands for! (American national standards institute)
The next thing to do is to check this website to verify that they are insured. The guys not on this list will have 100 excuses about why they're not on this list. Don't listen to them!