The OP is looking to hire someone to do tree work on the side on weekends (correct anything I say that is not accurate), and is asking if he is liable for injury or worse during that said work?
If the person doing the work does not have a policy that covers them in the case of an accident, that person can sue you for damages. If that person is covered under workman’s comp he cannot sue the homeowner for damages.
Here’s the deal. Your trying to get off on the cheap. I charge between $2200 and $2500 a day for a three person crew and equipment (I help manage a company and don’t own one). At the end of the day if we make out with 25% profit we’ve had a good day. We pay liability, comp, unemployment, health insurance, vacation, sick time, a living wage, 401k’s, equipment and gear replacement, and probably a bunch of other things I’m not thinking of right now. If someone gets hurt, your covered, we’re covered.
You pays your price and takes your chances. If your getting off on the cheap there’s usually a reason.
It will probably work out fine, unless it doesn’t, and then, well......