N.H. Rev. Stat. Ann. § 231:154

When any highway shall be laid out damages may be assessed to the abutting owners to provide for the maintenance or planting, from time to time, within the limits of such highway, of such shade and ornamental trees as may be necessary for the preservation and improvement of such highway. Damages may be assessed to abutting owners on any existing highway upon petition therefor, and such proceedings had as in the laying out of highways by selectmen to provide for the maintenance and planting from time to time, of such trees within the limits of such highways as may be necessary for the preservation and improvement of the same. When such damage shall be assessed and paid there shall be, in addition to the right of travel over such highway, a public easement to protect, preserve and renew the growth thereon for the purposes aforesaid.