回复：谢了，老瞎。 那我是否要回他信了？要回应该咋讲？

Generally speaking, it should be your responsibility.  Besides, there is no need to have a bitter neighbor over a few hundred dollars for the sidewalk repair.    But you could ask you neighbor to cut the roots near his sidewalk (or share some of the cost).  Or maybe cut down the entire tree if the tree, i.e maple has invasive root so that the roots won't damage the sidewalk again in the future.  You could ask you neighbor to shear the cost of cutting down the tree and removing the tree stump(if preferred).