If you dont know the hazard exists, you are not liable. If your neighbor sends you a certified letter making you aware of the hazard or in this case, if you get on the internet and tell all humanity you killed the tree so it will fall on your neighbors house, you are liable. If its an act of nature and you are unaware of the hazard, ypu would not be responsible, it would fall on the neighbors home owners insurance.
Pretty sure this is correct! If you have a leaning tree and the neighbor identifies it as a hazard to his property formally to you and you don't do anything and it is blown over by act of god and damages his property you are liable.