Well I can say that that is indeed the case in the UK: if a tree is in your garden you are required to take reasonable precautions. So if the tree is leaning by a large angle, is rotten or damaged then you are legally responsible for making it safe.
Furthermore, even if you have taken such precautions, if a tree is felled by an “act of god”, you are still responsible for some of the costs: namely the cost of removing the tree itself.