my neighbor cut the limbs off a tree that is half on both our property

It sounds like no one knows who owns the tree, or who planted between the first instance.  The basic rule is that you have the right to do whatever is appropriate to that portion of the treated overhangs your property.  Your neighbor owes you know duty of care, nor do you on your neighbor any duty of care with respect to this tree.  If you are concerned about the tree falling and damaging your house, you should trim the branches to the extent you deem necessary.  If the tree is dying, you might consider speaking to your neighbor about sharing the cost of hiring a tree expert to remove the tree safely and with minimal risk of damage to either house.