Usually in the UK we can erect a garden fence up to a height of 2m in the back of the property and we can cut back any overhanging branches as long as we return them to the owner. We also have tree preservation order that might restrict us? The local council would offer us advice on them.
In the USA as in the UK I suspect you still need to look into your rights Locally - look through your house deeds re your boundary and any covenants that may help or prohibit you? Also any local bylaws re errecting a fence, height, removal of trees etc. You have to do your homework locally I am afraid?
Tread carefully and know your rights first, as any damage trespass etc may end up costing you thousand of $

As Drc implies, in the UK, we are legally entitled to remove any growth from a neighbour's plants which overhangs our boundary - but only back far enough to keep your own boundary clear. Legally also, the cuttings do belong to the neighbour. I'm not sure whether that's also true in the States, but were you over here, I'd advise you to cut back to your property/boundary line, and then erect your fence along the boundary.

I would put up the fence you want but leaving the existing farm fence. If you need to remove any of the neighbors branches, encroaching on your land then do so - this is within your rights under NC law.

As you have kept the original fence then she will have zero comeback.

You will probably have to go to your city government for information. I looked up Haywood County on their website and couldn't find anything relevant. However, one of the cities did have on their website a place where you can request information. So Monday, call them up and ask what your rights are, and if they don't know, ask for a number of the correct city/county department. And if you start to feel you are being run around, call the Mayor's office and complain.

Samantics Bamboo! I said, not implied 'we can cut back any overhanging branches as long as we return them to the owner'. That sentence in no way implies I suggest touching anything past the boundary.

That's the second time you've used the word 'semantics' in a response to me, Drc - you have inferred a meaning not intended - I merely clarified by adding 'to the boundary line' instead of your 'overhanging branches', and used the term 'implied' because that's not quite what you said - although it was implied. Technically accurate on my part.

:))