As for authority, in Nebraska the state has given it to the "primary" cities (i.e. Lincoln and Omaha)
Lincoln Municipal Code wrote:8.46.010 Owner of Real Estate to Remove Weeds.
It shall be the duty of every owner of real estate in the city to cut and clear, or clear, such real
estate, together with one-half of the streets and alleys abutting thereon, of all weeds or worthless
vegetation whenever such weeds or worthless vegetation shall extend more than six inches above
the ground. Such weeds or worthless vegetation shall be cut so as not to extend more than six inches
above the ground. After cutting, all such weeds or worthless vegetation shall be immediately
removed from such real estate, or be burned upon the premises; provided, however, that if burned
upon the premises, a permit so to do shall first be obtained from the Air Pollution Control Officer
pursuant to Chapter 8.06 of this title. (Ord. 15569 §1; May 14, 1990: P.C. §8.80.010: Ord. 14936
§1; July 25, 1988: prior Ord. 11433 §1; August 18, 1975: Ord. 10926 §1; October 15, 1973: Ord.
3780 §1; November 18, 1940).
ANNOT.: The State has granted the cities of the primary class the right to
require the owners of real estate to destroy weeds and undesirable
vegetation and has made it the duty of the City to do so if the owner
So as I read it smaller towns and communities that are not "cities of the primary class" would not have the same authorization. Ah federalism, you have to love it...
The courts have ruled that the city cannot be held liable if city employees damage other plants in the process of destroy weeds on private property
Greenwood v. City of Lincoln, 156 Neb. 142, 55 N.W.2d 343 (1952)
Such activity is a governmental or public function. Municipal
employees in the performance of such a governmental function
cannot subject the City of Lincoln to liability for the destruction of
plaintiff's raspberry bushes.