@joshuabarksatlcs   You're correct that in order for neighbors policy to pay for resultant damage and tree removal you'd have to establish negligence on the neighbors part. Short of that, you clients Homeowners policy should respond.  I've always told clients to take photos and send removal or pruning demands to their neighbors by certified mail.  Sometimes it works... sometimes not... it certainly doesn't help the relationship with the family next door.... 

Overhanging trees have become a big problem nationally for homeowners and homeowner insurance companies.  While tree cover has always been an issue for insurance companies the weather has been more violent lately and insurance losses more costly.  With satellite and drone technology, it's easy and inexpensive for insurers  to inspect each home and evaluate the risk. I've seen lots of cancellations and non-renewals issued  for tree cover.  Normal insurance underwriting philosophy says eliminate the risks that will cost you the most in claims.... tree damage/removal is pretty high up the list today.