Kass - it can only be an old trick IF the tenants is in arrears cant it? And if someone owes you money do you have to trick it out of them?
Kass - you say often enough you are not a housing professional so is it wise to give advice on whether the HA will allow this structure to remain in situ?
As to why the HA may be against it and refuse and insist you take the tree house down. When you leave the property and say this is in 10 years time when your dauhter no longer needs or uses a tree house (or even sooner), then if you leave it at the property the landlord is liable for its safety. It would likely have no option but to dismantle it at a cost to themselves as they couldnt be guaranteed it was safe.
Hence to avoid that cost to them they can ask you to dismantle it. Personally I hope they dont but there are legitimate reasons why they might ask you to dismantle it.