Saturday, there was sustained winds, with some gusts topping off at 65 mph! So what do you get with 65 MPH wind - downed trees and branches! So if this was your neighbors tree that fell on YOUR garage, shed, pool or fences, it really is your claim on your insurance not your neighbors insurance! Your neighbor wasn't negligent or at fault here - the wind did the damage, or an act-of-God! Thus, through nobody's fault, your property was damaged and you now have an insurance claim.
Sorry to report this but that is how it works. Doesn't matter who owns the tree, it fell due to no one's fault!