Branches from native trees overhanging from a neighbouring property can be pruned back to the boundary as long as pruning does not exceed 10% of the tree canopy. Pruning works are to be carried out in accordance with acceptable industry standards. Pruning that exceeds 10% is subject to an application to Council.
You should discuss the pruning with your neighbour first and work should be carried out by a qualified arborist.
Offcuts are green waste and should be disposed in your green waste bin. In the interests of maintaining good neighbourly relations, offcuts are not to be thrown back onto the neighbouring property.
Council does not have the authority to enter into disputes between neighbours about trees. This is deemed a civil matter covered by NSW State Legislation Trees (Disputes Between Neighbours) Act 2006. You can seek legal advice from a solicitor, the Community Justice Centre, Chamber Magistrate at your Local Court, or the Land and Environment Court.