We often get phone calls regarding a neighbor's overgrown trees and whether they can legally cut the branches back. While there is no Arizona statute that addresses this issue, there is "common law" as well as some Arizona case law that addresses this issue. (Case law is composed of accepted tenets based on judgments and opinions from previous court cases.)
When it comes to encroachment of trees or other plants on adjoining property, the common law is derived from years of court decisions. These decisions have determined that one's property line extends upward into the sky and downward into the ground, like an invisible line.
It is acceptable then, under common law, that a property owner has the right to protect his or her property from encroachment.
In 1985, the Arizona Court of Appeals case Canon v. Dunn (145 Ariz. 115) said a landowner may cut off offending branches or roots at his property line, regardless of whether the plant is poisonous or non-poisonous.
You are not required to give notice for this self-help remedy. It is the adjoining owner's burden to "protect" himself from protruding roots or branches.
This does not give someone the right to trespass on the tree owner's land or airspace to accomplish this. In order to do that, you must have permission from the owner.
All this may not apply if the landowner with the problem trees is a municipality. In those cases, check with the city at (602) 262-6501 before cutting branches. Phoenix City Code 34-15 makes it unlawful to cut or trim or interfere with any trees or shrubbery on city property.