The size of the trunk really shouldn't be a factor.
If it's a 1 inch oak tree, protect it
If it's a 24 inch Norway Maple cut it, burn it, destroy it.........

It sounds like your woodland was originally a hardwood forest til
beneficial
fires were kept from killing off non native and harmful fast growing soft
woods which choke off the Oaks and Elms. That's probably a lot of your
undergrowth.

Thanks for the tips -- that's what I need, more direction on what to cut
down and what not to cut down.

I don't know much about hardwoods vs. softwoods ... are softwoods the faster
growing weed trees? Are there any desirable softwoods? I don't think we have
any Norway Maples in our woods ... I think I could easily identify a maple.
I always thought maples were a desirable tree, but according to you and some
others, the Norway Maple must be more of a nuisance. Anyway, I don't have to
worry about that one, but there are some shorter trees in our woods that
look different than all the others and that I can't identify, despite
consulting a couple tree identification books. I think I'll have our cities
arborist come out for a consultation.

Dave K.