Tell me about this "notch" in the tree. Was this notch put there on purpose by the owners of the property? Like they had planned to cut down said tree and didn't? And as a result this notch rotted and caused said tree to fall? There might be some negligance on their part IF this is the case. Might.
Otherwise you are basiclly on your own. You do have the right to remove any branches and roots of any tree in someone elses yard up to your property line. There is lots of case law on this.
ps give me some time to get used to your new name!