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Let me see if I can help. There is a considerable reluctance by LAs to remove trees where they have a TPO. That said in accordance with the Town and Country Planning (Trees)(Amendment)(England) Regulations 2008 anyone wishing to undertake works to protected trees due to suspected tree-related subsidence damage (to buildings, walls, conservatories, driveways, drains etc.) must submit a Standard Application Form with relevant technical evidence and they should either make good any necessary damage or potentially remove the tree or parts of it. You should contact your Local Council's Arboricultural Officer for more information
The principle would be that it is causing damage and there is nothing that they can do to stop the damage unless the tree is removed. They would want under such circumstances to replace the tree with a new tree as near as possible. So specifically trees are sometimes removed. It is not a case of there being caselaw on the matter (these things do not go to court where decisions are recorded) but I am aware that it happens from experience.