Taken some more photos and found an arborist who is coming over to confirm what I already know and write up a report. Found some white / brown gunk which felt a lot like sap which I presume is poision on the ridges of the trees. This was on the two main trees. One that has died and one that is dying. Then all the trees have branches that have been broken off yet other lemonwoods that are more exposed we planted at the same time on a different boundry are fine. Plus some of the green foliage underneath the trees, where the trees that have died or are dieing there is no foliage, or dieing weeds. Yet right next door on the ok trees the weeds are growing fine.

Pretty conclusive IMHO.

All those branches coming out at that point of the tree doesn't look all that healthy, and the liquid looks  like sap? Maybe testing of the liquid may tell you for sure.

And even if does turn out to be poison, you would still have to prove to a reasonable standard that it was your neighbour who did it before the Police could do something about a willful damage charge. If they deny everything, and you have no evidence that it was them, I doubt anything could be done.

BarTender:

 

Taken some more photos and found an arborist who is coming over to confirm what I already know and write up a report.

 

Personally, it's beyond me why you keep bothering to waste everyone else's valuable time and effort if you are simply determined to close your mind to any other rational possibilities. There have been posters in this thread who (on the face of things) have some knowledge about these matters that are very strongly suggesting to you to be more open-minded, yet you insist on closing your mind. 

As someone who has experience advising people on neighbourhood disputes in a community law centre and  who's prosecuted a couple of so called "ordinary blokes" who allowed their emotions to get the better of them in neighbourhood disputes, I can very strongly advise you that you're giving off a lot of the signs that those types of guys did before they got themselves in trouble. Your attitude is potentially very unfair to your neighbour and is quite liable to make you do something silly. Stop it before you make an ass of yourself or worse.

It is sad when you see trees outside people's boundaries which looked absolutely healthy and strong and perhaps few decades old being put down. When you look at the cut and see no rot you are asking why? When you look at the property behind it and see it is for sale or another property undergoing extensive renovation - you are starting getting ideas ...

You can not return those beautiful trees on Matai Road, Greenlane, Auckland. Unfortunately.

I can understand as to why you believe these trees may have been poisoned. However we will all never know unless you get tests of the earth and trees done. As someone who has spent many years on a farm as a horticulturist it doesn't look like poison has caused this damage. Have you possibly poured something in the grass there, Maybe someone else did a few years ago before you and the roots have only just got that far. Waste oil and other chemicals can cause that. I think someone asked if there were drainage pipes there or one of those sewage systems that break down the solids then spread it through the garden? Are there?

Just remember you have to live next to your neighbors and they have to live next to you. I think that says enough. 
Its easier to live with 3 dead trees than it is with constant noise control and what ever else they can inconvenience you with. People can and will be nasty. Just dont make that person you.