An individual cannot, under any circumstances, just remove lop or trim a tree on a property in separate ownership. Councils do not generally get involved in negotiations between neighbours regarding trees. It is up to you to talk to your neighbour and work out a mutually acceptable solution.
If your neighbour decides they do want to remove a tree they will need to liaise with council to determine if a planning permit is required.
If the land is Council owned, you will need to submit a Service Request Form stating where the tree is and the issue. A council representative will inspect the tree and if a decision is made to remove it, there may be a public notification period, which allows members of the public the opportunity to lodge a representation.
If the tree is on Crown land, you will need to contact the Department of Primary Industries, Water and Environment to obtain consent. The removal of trees on Crown land may also require a planning permit