So I had a similar situation at my last house. My b**** of a neighbor sent me a certified letter making note of a huge tree in my front yard. None of it was dead, or dying, or even looked bad. It provided tons of shade over my house, and looked beautiful. The letter stated that any damage coming from the tree or any parts of it falling (on their bushes, not their house) would be my responsibility. I laughed, didn't do a thing - and have since moved. The buyers of the house ended up having the tree cut down at their expense due to the ongoing requests, I feel bad for them.