If the tree is not on Housing or Council land, it is highy likely that the Housing Officer does not know who the tree belongs to. The complainant should check with the land registry.
Before cutting anything back, you also need to check that the tree is not covered by a Tree Preservation order.
Also, how has the tree request been made? Has it just been thrown into the conversation whilst the housing officer has been on site, or has a formal question been asked? I think a lot more detail needs to be given on the matter prior to any complaint being made.