In NSW the law states that you are permitted to trim any part of a tree that encroaches over your boundary. You do it at your expense not the tree owners will expense.
You are not permitted unless the tree owner agrees to go onto their property to do the trimming.
Having said that local ounces also restrict the amount that can be trimmed in one go from a tree (in my country il area its 10%) and the coun is requires a registered arborist to do the work.
Unless the neighbour gets a court order ,ordering you to pay, they are required to pay.