If you go to England, you're better off going to the New Forest in Hampshire. Despite the name, it was established during the Norman period and at least feels like a substantial tract of older woodland. (Tiny by North American standards of course.)

Forest was more of a legal term, and tended to be focused on deer than trees.

Things with 'new' in the name tend to age quite rapidly. See also Oxford's New College.

Or the Pont-neuf ("new bridge") in Paris being the oldest one in the city.