We have a magnolia in our garden which shades our neighbour's patio a bit - a couple of years ago (just after we moved in) they asked us if they could pay for someone to prune it for us so it didn't block so much sun. We said fine but we couldn't afford to contribute to the cost. They did it and it was a really bad job and now the tree is a horrible shape. And since then they have been chopping off all the branches that go past the fence onto their side and now it looks even worse - I loved the tree when we moved in and now its just all lopsided with bare branches pointing out all over the place.
I suspect its the case that you are allowed to chop off anything that hangs over into your garden, but what happens if all the hacking at it is damaging the tree?