For the most part, the answer is yes. This is not limited to your main residence and would include a detached garage, a guest house, shed, or anything that is not specifically excluded in your policy. This includes any contents as well. The tree can be felled by natural disaster (windstorm, lightning, etc), but if your tree came down due to neglect, that may not be covered.
Whether you own the tree or not doesn’t matter as far as filing a claim goes. Especially in cases of a windstorm where branches, bushes and whole trees can be projected from great distances, ownership isn’t something that could be readily assessed.
Standard home insurance policies also provide coverage for damage to trees and shrubs due to fire, lightning, explosion, theft, aircraft, vehicles not owned by the resident, vandalism and malicious mischief. Coverage is generally limited to up to five percent of the amount of insurance on the structure of the house. Generally, most insurers will limit the coverage to about $500 for any one tree, shrub or plant.
Of course, if the tree was located on your neighbor’s property, your insurance company may attempt to collect from your neighbor’s insurance company, especially if the tree fell due to poor maintenance. You may be reimbursed for any deductible. If a tree fell on your property but did not damage anything, it is unlikely you will receive anything to pay from removal. Exceptions could be if it is blocking your driveway or some other egress.
Why not talk with your insurance professional to go over all of your insurance policies and see if it is time to update your protections? Make sure you are covered for everything you think you are, and some things you have not thought about at all.