Sorry to hear about the tree down! Home insurance does cover some tree removal fees, but it depends on the cause of the fall and the location of the tree.
Generally speaking, home insurance covers damage caused to your property. So, if the tree did not fall on any structures or buildings, then removal of the debris may not be covered.
Most policies have a limit for tree removal costs, usually around 5% of the policy limit amount. Check your policy carefully before you decide to submit a claim.