It’s up to neighbours to work out between themselves who will apply and pay for a tree permit. Council doesn’t deal with disputes about trees between neighbours.
If you have a dispute with your neighbour about a tree, you may need to seek help through private legal action. You may also need to get legal advice if a neighbour’s tree has caused damage to you or your property.
If the matter can’t be resolved through mediation, you could consider lodging an application with the Land and Environment Court under the Trees (Disputes Between Neighbours) Act 2006.