Actually home owner's insurance will pay for a tree that falls on your house, even from another yard. I know this because our oak in the backyard broke off during some straight line winds a few years ago during tornado season (<50mph) and it fell down and brushed against our neighbour's house (different one) and it damaged her shingles and gutters. I was going to call our insurance company but she told me that hers is the one responsible. I called ours to be sure and they said the same thing.
What I can remember from my Risk Management class.
Your tree falls on your house, your HO pays. Your neighbor's tree falls on your house, their insurance won't pay, yours does. Your tree falls on neighbor's house, their insurance pays.