I have always believed a strong selling point for being a real estate lawyer is we get to answer such burning questions as, "Who owns the sidewalk?" or, "Can I cut down my neighbor's tree branches?" It is the latter of these two questions the Vermont Supreme Court addressed recently in Alvarez v. Katz, 2015 VT 86 (VT. 2015).
And the answer? I bet you'd never guess: It depends.
If your common property line passes through the trunk of the tree, the tree is a "line tree," you own the tree in common with your neighbor, and neither you nor your neighbor may destroy the tree by cutting the portion lying on one or the other side of the property line.
However, if the trunk of the tree lies completely within your neighbor's property but the branches and roots overhang or intrude upon your property, you may freely cut those branches and roots, even if to do so would be to destroy the tree. For an example, take a look at the scene from Door to Door (a movie I highly recommend beyond its utility illustrating land-use issues):
One caveat: this is a case from Vermont, and other states are free to establish their own rules. In Arizona, it might be important whether the branches and roots are actually causing you injury. Consult a Tucson real estate lawyer who can help determine your rights to cut down tree branches and roots intruding on your property before you bring out the chainsaw.