The answer is yes and no, as usual with insurance it depends on the circumstances. Let’s say a tree branch falls through your roof and causes damage to the roof and the rooms below, the resulting damage is obviously covered, and because there is damage the removal of the tree branch is covered too. However, if the tree branch falls on the roof but, causes no damage, there is no coverage to remove the tree branch on your policy, or the coverage is very limited to maybe $500 maximum. You can buy tree removal coverage by endorsement on some homeowners policies, ask your agent.
If a tree falls in your yard, there is no coverage to remove the tree but, if it falls in your driveway it would be blocking your access and therefore would be covered.
If your tree falls on your neighbors house or car, your policy will not pay for the damage, your neighbor’s insurance policy must pay for the damage. However, if the tree was neglected and appeared unstable prior to the fall the neighbor’s insurance company might file a claim against your insurance policy.
Lastly, if a tree falls on a car, the auto insurance policy for that car is the only policy that will provide coverage. The coverage that applies is called “comprehensive coverage” and it will pay to have the tree removed from the car. If you did not elect to purchase comprehensive coverage then there is no coverage for the damage or tree removal.