Mmmm, it isn't "wooded" preserves that make for a correctly-tagged leisure=nature_preserve.  The "landcover" could be woods, swampy or even desert (say certain kinds of sagebrush and a species of roadrunner is part of what is being "preserved").

It is the function of both "protecting natural biology" AND that humans are allowed (usually to do very low-impact activities, like read signs, observe with binoculars, hike only if strictly staying upon trails...) which makes the unique "dual-use" aspects of a leisure=nature_reserve.  Yes, the natural biology is "preserved" here, yet humans might also recreate (leisure), but only if low-impact.

Semantics like "wood" are best left to natural=* tags.  We express TWO DIFFERENT semantics with tagging both leisure=nature_reserve AND natural=wood, for example.  The former does NOT imply the latter...that's why it's appropriate to use two tags to tag the two meanings.