What to do when a city tree sprain your house? ...
Answers: Call your insurance company for advice. They should be capable of point you in the right direction. You may stipulation to file a complaint beside the city to start with.
Trees, or other things falling on and hurtful your home, are considered to be "an act of God." Fault is usually not assigned. Your own homeowner's insurance company should cover the injury. Take pictures and call them to unequivocal your claim.