First, this situation is referred to as “abatement of a nuisance":
the removal or termination or destruction of something that has been
found to be a nuisance.
"Suppose a tree trunk stands completely on one person's land, the
branches project over another person's land, and the tree roots
extends into the soil of another...the other property owner may cut
the branches and remove them if they are a nuisance...The adjacent
owner is regarded simply as tolerating a nuisance. In addition, the
mere existence of a nuisance for any length of time will not allow to
create a perspective right. The adjoining owner may cut off the
projected branches if they're a nuisance."
Source: Blackstone Paralegal Studies: Real Property, Part II
====================
If the roots or branches of your neighbor’s tree encroach on your
land, you can cut them back to the boundary line. If you do not 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
neighbor’s property.
Nor may you create any other problems for your neighbor. 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 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 neighbor’s land
belongs to them, and they will be liable for any damage it causes.
However, if the tree was planted on the boundary, you are probably a
co-owner. If your neighbor does not agree to have the problem
resolved, you can apply to a district court for an order for removal
or trimming.
Who pays? 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 neighbor’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 neighbor’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 neighbor’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 neighbor’s enjoyment of their own properties.
Local councils are generally reluctant to become involved in
neighborhood 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 Historic Places Act. 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 arborculturists.
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 authorization.
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.
If a tree owner and an aggrieved neighbor can't agree on what to do,
several courses of action are open.
Mediation and arbitration - Both mediators and arbitrators are
available to help resolve a dispute. However, neither party can be
forced to take part in either of these processes.
A mediator will help you negotiate a solution to the dispute. An
arbitrator will impose a solution. Mediation is less formal and
usually less expensive, but cannot be enforced by a court unless you
have included enforcement procedures in your agreement. An arbitrated
settlement is backed by the courts.
Before you start, you should work out the likely costs. Mediation and
arbitration are charged on a time basis, and both parties are expected
to pay an equal share, unless another agreement is reached.
Disputes Tribunals can hear claims for damages to property for amounts
up to $7,500 (or $12,000 if the parties agree). Typical examples are
claims for damage to drains, driveways, foundations and fences.
However, generally a tribunal referee will not be able to hear claims
when the dispute is over loss of light, sunshine or views, or involves
removal or trimming of the tree. In the latter case, a referee can try
to help the two sides reach agreement. However, if your neighbor
decides to ignore this, you will have to go to the district court to
try to get the problem resolved.
District Court - Claims for more than $7,500, or that involve the loss
of light, sunshine or views, or that involve the removal or trimming
of trees, can be taken to a district court. The court can award
monetary compensation for damage caused by a tree. It can also order
that a tree be removed or trimmed. Claims through the District Court
will almost certainly require the help of a lawyer and can be
expensive.
Source: Consumer On-Line
http://www.consumer.org.nz/topic.as...&title=Your rights&contenttype=summary&bhcp=1
Sadly, they have to cut sweet cherry tree off to extend from small garage to bigger one... 
. It was funny that my wife saw what I did and she said hey that look funny!! I told her so what! As long as cranapple falls into THEIR property, not OURS!!. So at least I don't have to worry picking up fallen cranapples on my property.
They leave messy on their yard!! Long grass, sick tree, bee hives on their brand new house! 
