The Kaipara District Council generally does not control the effects of nuisance trees on private property. Instead, Council defers to the Property Law Act.
The Property Law Act 2007 says property owners are responsible for any nuisance or damage their trees cause to neighbours, even if the trees were planted before they bought the property. If a neighbour's tree is causing a nuisance, you have a range of options available, from a friendly chat through to a court order.
You should first approach your neighbour and try to explain the problem as you see it, as you might be able to reach an amicable solution.
If you do seek a court order to have a tree trimmed or removed on your neighbour's property, in most circumstances you will be required to meet the cost of the work. You will also have to provide evidence of the nuisance and convince the court of the merits of the action you seek.
Kaipara and Northland have a number of well-qualified arborists who can provide good advice and services to deal with tree issues.