I found this city ordinance that obligates property owners to trim trees, shrubs, and other vegetation from overhanging the sidewalk and other portions of the public right-of-way, but I would have to say that this is certainly not being enforced or communicated to the public with any regularity.
Sec. 701-6. Duties of property owner.
(a) The owner of any private property in Marion County which borders or lies adjacent to any public street, alley, right-of-way, place or park and upon which any trees or flora may be standing shall trim or cause such trees or flora to be trimmed, either at the property line, or to a clear height of at least eight (8) feet above the surface of any abutting right-of-way or place, and fourteen (14) feet above any public street or alley. All branches or parts thereof which overhang any portion of a public street, alley, right-of-way, or place, or which obstruct or interfere with the passage of light from any street lighting system, shall be trimmed or cut. No person shall plant or maintain any tree or flora so close to any property line as to obstruct the vision or free passage of pedestrians or motorists along the streets or public right-of-way. The consolidated city or its contractual agent or agents may enter upon private property to do such cutting or trimming as may be necessary to remove any offending and obstructing tree or flora that is prohibited by the provisions hereof.
(b) An owner shall, and the consolidated city may, remove from flora all dead, decayed, broken or dangerous limbs, branches or parts thereof or any that overhang or are close to any public street, alley, right-of-way, or place, and, when any such flora is dead, the owner shall completely remove the same, or after notice to and failure of the owner so to do, the department or its lawful agents may cause such removal and charge the cost thereof to the owner in accordance with the procedures hereinafter provided.
(Code 1975, § 30-6)