Sounds like they are trying it on. What is the tree?
Is it evergreen and is there another evergreen tree in the garden which would bring it within the letter of the high hedges act? ie a line of two or more evergreens.

Time to start talking to the neighbour, invite round for a drink. At the end of the day they would not seem to have any legal claims over your tree. Very unfortunately, there is no such thing as a 'right to light' and a single tree does not come under the 2m ruling. Talk to them and you might find that you have a lot in common.

Is there a town ordinance on tree height? Then there's the other question of being a good neighbor. You should also go around to their side to see how your tree is affecting them. How much do you value the tree? Is it worth all this ruckus?

Not in the UK Bathgate... However I agree Jinvin needs to go into neighbour's garden and see what effect his/her tree is having on their garden, could well be blocking light.

Even if it is though they cannot compel you to do anything about it. What action you take really depends on whether you want friendly relationships with your neighbour. Depends how badly you want the tree and how much light it is actually blocking. My neighbour once killed a clematis montana that was growing along a post and wire fence about three feet high because, she said, it was shading her border - and there wasn't much growing there anyway. But I just sighed and left it at that because she was a nice lady and we got on well.