Thin to improve forest health and diversity

I purchased some property 3 years ago, 37 acres of woods, and it needs to be thinned. It has too many trees, mostly oaks and pines, and too much undergrowth, mostly maples. There appears to be canker on some oaks. After I identify the trees to be taken down, should they be removed from the woods also? There are also many standing dead trees. I assume yes but want an experts advice. By the way, this property is surrounded by Mark Twain National Forest.
Thanks in advance.

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