Again, it doesn’t matter if the fall tree was on your property or your neighbors. The original location is unimportant, as the insurer will only care about where it ended up. If it’s your property that was damaged, it’s up to you file the claim. You will be responsible for the deductible, but if the tree was your neighbor’s, you may be reimbursed for that portion.
If you live in an area susceptible to hurricanes, the deductible could be much larger, since it is estimated as a part of the home’s insured value. Take an inventory of your yard and see if there are any unhealthy trees: ones with split trunks, exposed roots, etc. A policy may cover damage caused by fallen trees, but performing proper maintenance is important. Your insurance provider may challenge a claim if the trees weren’t well kept.