I'm not sure about Ethel's answer. There's a strong case for saying that ethically or morally you should pay (and, clearly, if you decided to have the tree removed, it would be your responsibility to pay and the cost would be far greater than just pruning it), but I would question whether there is a legal right for the neighbour to recover the cost they have incurred from you. It seems to me that it could well be argued that, if they want to be able to do this, they should seek your agreement in advance to the estimates of cost which they will have obtained, and give you the opportunity to get estimates of your own.