Re: Butternut Trees
Butternut trees (Juglans cinerea) are very closely related to black Walnut trees (Juglans juglans). It would be difficult to tell the seedlings apart. I found this description of a differing characteristic between the two that might be helpful: The underside leaflets of butternut are densely covered with stellate hairs, while in black walnut leaflet hairs are almost inconspicuous.
If they are butternut trees they are probably from nuts that have been lying dormant for years. The new clearing has given them enough light to germinate and grow; full sun being something that butternut trees insist on. I would consider keeping some of them as this is a somewhat rare tree that produces tasty nuts and beautiful wood that is valuable for furniture making. There is a canker disease (Sirococcus clavigignenti-juglandocearum) that has wiped out most of the wild butternut trees so this species could use all the help it can get!
That being said, if you continue to pull the seedlings, they should stop coming up, since the parent tree was probably one of those that fell down.