Cutting & Trimming
If the roots or branches of your neighbour's tree encroach on your land, you can cut them back to the boundary line. In law, this is called "abatement". If you don't want to do this yourself you can ask a district court for an order for the trimming or even removal of the tree.
However, if the tree is not causing harm or loss of enjoyment, abatement may be your only remedy. If you do choose this option, you must do no more than is necessary to abate the nuisance. No unnecessary damage should result, and you should not trespass on your neighbour's property.
Nor may you create any other problems for your neighbour. You must not poison the roots or spray the tree with herbicide, as the consequences would extend beyond your property. If you are cutting out part of the tree's roots, take care not to undermine the stability of the tree or the ground around it.
Cuttings and fruit belong to the tree owner. You can put them back on their property, taking care not to cause any damage, or ask for them to be removed.
If the trunk of the tree extends over the boundary, this does not give you the right to chop it down. A tree planted on your neighbour's land belongs to them, and they will be liable for any damage it causes.
But if the tree was planted on the boundary, you are probably a co-owner. If your neighbour does not agree to have the problem resolved, you can apply to a district court for an order for removal or trimming.
If you have incurred costs in cutting back the roots and branches on your side of the boundary, you probably will not be able to claim them back from the tree owner.
But if the roots of your neighbour's tree have damaged your drains or a branch falls on your house, they will probably have to pay. Even if the damage results from forces outside your neighbour's control, they may still be liable if they could have been expected to know the tree was unsafe, and did not take reasonable steps to make it safe.
This means they will have to pay the costs of fixing up the problem as well as any compensation that may be due.
Even if your neighbour's tree has caused no damage, but is simply being a nuisance, perhaps by blocking sun or light, they may still be liable for the cost of getting the nuisance resolved. This is because tree owners should take reasonable steps to stop the trees interfering with their neighbour's enjoyment of their own properties.