Local councils are generally reluctant to become involved in neighbourhood disputes about trees.
However, many trees are protected. Classification of a protected tree will vary among councils and may include specimen trees above a certain height, native vegetation, or even "blanket protection" of all trees in your area. Before you start to chop down all or part of a large tree, check with the council whether you need special permission.
Other forms of tree protection include the listing of significant trees in a district plan, heritage orders under the Resource Management Act and voluntary protection under the Heritage Covenant provisions of the Heritage New Zealand Pouhere Taonga Act 2014. There are substantial fines for ignoring some of these protections. Your local council will be able to supply you with details of its policy.
Some councils will also supply information on tree care and will give names of recommended arboriculturists.
If your tree is creating problems near a road or public land, the council can issue a notice ordering you to remove or trim it. This might happen if the tree is damaging roads, drains or other public amenities, or if it obstructs traffic or the view of road traffic. Several other statutory authorities also have this right.
If you want to challenge the council's view you can apply to the district court to have the notice set aside. But you will need to be quick: in some situations you will only have 10 days in which to do this.
If you simply ignore the notice, the council can enter your property and carry out the work itself. You will have to bear the cost and may also be fined. In an emergency where there is imminent danger to life, property or roading, the council can do this at your cost with only verbal notice being provided beforehand. But it cannot do more than is necessary to prevent danger.
The council must also look after its own trees according to the same basic rules as everyone else. Most councils have written policies covering this, which vary from council to council. The question of who pays for what also varies.
The first step is to let the council know there is a problem. If you are not happy with the response, you can proceed as if the council were a private landowner: perhaps by cutting back the offending roots and branches, trying to get the council into mediation or even starting legal proceedings.
If you want to plant trees or shrubs on council land, you must get permission first. It is an offence to do this without authorisation. Illegal plantings can interfere with drains or public works, or they may be considered inappropriate for a particular environment. It is also an offence to remove or damage trees or shrubs growing on council reserves, except within the normal scope of abatement.