It is fairly easy to answer the question -- who would be sued, the tree company or the person who hired the work. They both would be named defendants in the lawsuit. In the course of discovery, it might become clear that only one or the other, or perhaps neither, was actually liable. If the parties do not reach a settlement, in the end liability may be determined by a judge, jury, or arbitrator. One should have a contract including a clause that requires the person who hires the work to protect, defend, and indemnify the tree company in the event of a claim for cutting the tree, and should describe with particularity the tree to be cut to ensure the property owner does not later say it was a different tree that was to be cut. The language used in the contract should be drafted by an attorney since the law and court practices may vary from place to place.