The owner of the tree is responsible for maintaining it in good order. In the case of tenants or leaseholders, it is usually the landlord who is responsible unless the tenancy agreement or lease specifically says otherwise. The responsibility for maintaining a tree does not mean you must actively do something to look after it but if someone is injured or property is damaged because you didn’t look after it a liable case could be brought against the owner. This is called negligence.
When it’s obvious where the tree is growing ownership can be assessed easily. However, if a tree is growing adjacent to or spanning boundaries ownership detection can be problematic. To find out who owns the tree you need to establish where the boundary line runs and on which side the tree first grew or was planted. Neither of these are easy to find out especially if the tree is mature. Your property deeds will sometimes say if a tree is on your land but otherwise you may have to come to an amicable agreement with your neighbour. Ultimately, if you are willing to take the matter far enough, a Court would decide for you.
Another thing to remember is that all trees are owned by somebody. There is no such thing as “no man’s land” – all land and therefore all trees are owned by somebody.