This varies widely across the country, so comparing one community to another is rarely an apples-to-apples proposition. Over the decades since Dutch elm disease swept across the country necessitating extensive removals, trends are toward more formal programs and dedicated funding. Some communities are even beginning to regard trees as capital assets, thereby moving funding out of the operating budget and aligning it with grey infrastructure investment in sidewalks, curbs, roadways, etc.
In Wyoming, urban forestry activities are funded through the operating budget along with snow management, seasonal leaf and Christmas tree pick-up, road repairs, facilities maintenance, parks and recreation maintenance, waterworks, and more.
The Ohio Revised Code permits an assessment based on property value or feet of frontage to be used solely for urban forest management activities. As of 2019, the assessment in Cincinnati is $0.19 per front foot; in Toledo and Cleveland, it is higher. At this moment, Wyoming is not using or considering such an assessment.
Municipal Tree Care and Management in the United States: A 2014 Urban and Community Forestry Census of Tree Activities — full report is ~78 pages; see Section Two: Tree Care Funding.
Trees as Capital Assets — ~2000 words. Circa 2003; represents a change in thinking from trees as an amenity to trees as assets.