Neighbor’s tree is going to fall a second time

I’ve already had one section of my neighbor’s tree fall on my fence and out building and had to file a claim. When our arborist gave us an estimate the first time he wrote “hazardous tree” and suggested we send the neighbor a registered letter stating that with a copy of the estimate. The neighbor seemed like he was going to take care of it so we never sent anything, that was six months ago and he’s done nothing.

I got to thinking, if my insurance company knows I sent a letter saying it’s hazardous do they bail out and not cover me when it does fall? Will the neighbor’s insurance not cover anything either since he knows it’s hazardous and then I’m 100% on the hook for everything?