perhaps closer photos of this part, and bud will help other to identify that, since it was years and years ago since I see any Fagus or Carpinus.
I trim your pic, seems the leaves are doubly serrated
from "A key to discerning American beech, Hop Hornbeam (Ostraya), Hornbeam
(Carpinus) and Birch. P.M. Eckel, 2013. Guide and illustrations."
Most carpinus I ever saw has double serrated leaves edge
Most Fagus I ever saw has single serrated leaves edge
Bark and trunk can be very variable within one species, let alone within genus, so I will not exclude possibilities just with one characteristic, also if you found fruits or floral part that will be very helpful, in most cases more helpful compares to trunk, bark or leaves