Who is responsible for damages to personal property from a fallen tree on someone elses property ?

Although far more information is needed to give you a complete answer, the general rule of thumb depends on how the tree fell. If it was being cut down by the owner of the property or a company then they would be responsible for the damage. However if the tree was knocked over by a force of nature (winds, lightening, ect) and the tree was determined to be healthy before the fall then no one is clearly at fault and insurance companies offer the best option for solving the issue.  If it can be proven that the tree was dead and that the property owner had knowledge of the problem but elected to ignore it, that property owner could be held liable for all the damage. For your direct issue you should talk to your insurance company or local counsel and determine your course of action.