I live next door to a council property. In the front of their garden is a seriously large tree. It's massive. The branches of this tall tree overhang into my garden. The council has proposed removing some of the lower branches that overhang my garden, but not the higher branches. The higher branches will still be in my 'airspace' (for want of a better word), and I will still have to deal with all the fallen leaves and so on.
So, I think that answer is 'no' but just in case - do the council have to remove every branch that overhangs my property? Or is it just my tough luck?