Laws about property owners' rights and disputes over trees, hedges and other greenery on or near property lines vary from state to state and even county to county, but there are some underlying commonalities in real estate laws that dictate how neighbors deal with issues regarding property lines and any disputes that may arise. In the case of trees on property lines, an attempt at discussion and compromise, involving respect for all parties concerned, should be made between neighbors before any legal action is taken.
Determine whether the tree is located exactly on a property line or whether the trunk has been planted on one owner's property (often the latter is actually the case). According to Larry Tankersley of the University of Tennessee, this determination can be made by figuring out who planted it (you, your neighbor, or a previous owner of either property) and where the whole of the tree trunk lies. The side of the property line on which the tree lies determines whose responsibility the tree is and who has the power over whether the tree stays or goes; however, if the tree literally DIVIDES the properties and is considered a "boundary tree," it is the mutual responsibility of both property owners to decide the fate of such trees, and any decisions must be agreed upon by both owners.
Discuss what steps you and the other property owner wish to take next, based on whose property the tree is on (or if it is literally a boundary tree and is therefore on both of your properties). If the property owner(s) responsible for the tree wish to get rid of it or cut it back for aesthetic reasons or the health of the tree, they may legally due so, but in the interest of your future relationship with one another as neighbors, both parties should try to make this decision amicably and with each other's wishes or concerns in mind.
Notify your neighbor (if the tree trunk is on your property and you decide to cut the tree in question down) or anyone else in the vicinity of the tree when you are planning to cut the tree to avoid accidents and/or hard feelings. If you and your neighbor come to a mutual decision to cut a boundary tree, it is a good idea to notify other neighbors who might be effected as a gesture of neighborly goodwill. If the tree is a boundary tree and your and the other property owner responsible for it cannot agree on whether or not to cut it down, you will need to pursue legal arbitration or litigation at either the local courthouse or town/city council.