The fence was there before the tree was planted. To repair or replace the fence the tree would need to be removed or the fence would have to be built off to one side or the other of the property line.
Estate Planning Attorney
I once had a yard maintenance company even out my lawn. Thy used a pick axe to take out some of the tree roots that were breaking the surface. Didn't hurt the tree at all. I would call someone that knows how to do this without killing the tree - then tell your neighbor that you will pay for the service to reduce this surface root to level out the fence - give them the info on who is going to do the work so they can be assured that it won't kill the tree. Hopefully they will agree to this and your problem will be solved.
You can estimate the size of a tree's root system by looking at it's canopy. As a rule of thumb the root system is bigger then the canopy. A tree can usually survive the removal or reduction of a single root just fine. But you also need to know something about the type of tree as not all trees are hardy. Trees tend to send roots to the surface in search of water when they are not getting deep watered. So you also might want to see how to set up something that feeds water more deeply to the tree to help prevent future problems if that is a possibility.
Another solution might be to adjust that section of fence so it just sits higher and is not being moved by the tree root. It is usually better to work with your neightbor to find a creative solution that you both can live with then run off to court suing people. http://www.portlandlegalservices.com
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