I knew that working as a risk manager for 35 years and having a tree fall on our house in 2015 would pay off someday. In a nutshell, if an ordinary healthy tree on your property falls onto your neighbor's house, it is your neighbor's homeowners insurance that will cover the claim. If you have knowledge as to a dangerous tree on your property that you did nothing about and it falls onto your neighbor's house, their homeowners insurance can subrogate against you and your insurance, so that you or your insurance company pays the neighbor's claim.
When the neighbor's tree fell on our house, leading to 10 months of repair and a $ 68,000 homeowner's claim, I had the tree service come and remove the tree from the roof. My neighbor was still having the house constructed, so the property belonged to the developer. I had the arborist go through and mark each tree on the neighbor's property adjacent to our house that could be considered dangerous. I then sent a certified letter to the developer enclosing the arborist report and asking him to remediate the dangerous trees. I advised that failure to do so would result in my insurance company subrogating against him should any of the dangerous trees fell on my house. The developer had his own arborist come out and look at the property; that arborist agreed as to the dangerous trees and the developer paid to have them taken down.
So the moral of the story, is if you wish to avoid a possible subrogation claim against you for a known dangerous tree falling on your neighbor, you should have them removed. You should also talk to your insurance agent or broker, but I would be surprised if their advice differed markedly from my advice here.