If it's a third party claim (you making the claim against his insurance) there shouldn't be any deductible. Sounds like there may be a miscommunication there or your neighbor doesn't want to make the claim.
From what you've said, the liability is pretty clear against the tree trimmer. He's the ultimate target. Commercial liability policies DO have a deductible sometimes, so maybe that's what your neighbor is talking about, but a deductible doesn't absolve anyone of responsibility.