The person(s) whose property is damaged would file a claim with his or hers insurance company, no matter who owns the tree. A typical homeowner's policy will pay to repair the damage to the structure and contents subject to the policy provisions. Your insurer will pay for removal of the tree; however, the cost for removal of the tree is typically limited per tree or per event. This Is typically referred to as debris removal coverage in your homeowner's policy.
Also keep in mind that the typical homeowner's policy will not provide coverage to replace any tree that was damaged/destroyed by wind. Debris removal coverage is also typically available for those circumstances when a tree falls and blocks the homeowner's driveway or handicap entrance/exit. If your insurer believes that any party (neighbor or city) has any liability due to the property damage, the insurer will subrogate against the other parties.