You can take some action. For instance, if overhanging limbs are a problem, you can trim them back to your property line. Yet, from the sound of things that may not solve the problem since it appears that the entire tree is in danger of falling down. What you might be able to do is to get help from a utility company if the tree is threatening to bring down electric, cable or phone lines. Also, city/county governments often step in and take action to make an owner take care of dangerous conditions, such as hazardous trees on their property. If that doesn't work, you can always sue your neighbor. The legal theory would be "nuisance", since that the dead tree interferes with the use and enjoyment of your property. Accordingly, you can ask the court to order the owner to remove the tree. At this point, you should consult directly with real estate attorney for further advice.