To classify it in the range that includes urban and suburban, a forested area beyond those sections would still be classified as rural.
Note that you could have a densely wooded area within the boundaries of an otherwise built up city. It might be considered an urban forest. You could also have a totally denuded area, far from any city (such as a strip mine) which would be considered rural.
So, urban vs. suburban vs. rural basically indicates the locality's population density, rather than the specific amount of natural vegetation.
EDIT — I also like @deadrat's answer of "sylvan", but wanted to show a horrible example of how that word is often misappropriated by development marketers, in places that are clearly suburban: