neighbors split tree limb hanging over my house

[Jeanette Vanden Bosch]

My neighbor has an investment property he rents out. My investment property is my summer home. He has a giant tree very near the property line. One of the limbs of this tree is split and hangs directly over our back bedroom. I sent him a letter along with pictures of the tree. His response to me was at first that we should cut them? Now he wants us to get 3 estimates and they will split the cost with us to trim the limb.

In addition to the limb the roots have cracked our driveway and are lifting part of his drive so that the water flow comes to my drive and my house.

My question is why am I responsible at all...having been notified there is a problem with his tree limb why should I have to do the leg work and pay half to enhance his property's safety and value?