Sorry about your fallen tree! We’re so glad to hear that it missed your home. Fallen trees are covered by home insurance only in some situations.
Since the tree didn’t damage your house or a detached structure, the tree removal is probably not covered, meaning you’ll have to pay for tree removal out-of-pocket.
If you want to try filing a claim anyway, expect a claims adjuster to come out to your property to assess the situation. If they find that the tree was already rotting, your coverage will be denied because rotten trees are considered preventable sources of damage.