Hi CH69:
You can help elicit excellent and educated advice by providing a more full picture of information about your plant.
First, where are you living and growing this plant? That makes a huge difference in giving proper advice about pruning and other sorts of care.
Second, since this is your plant, TAKE MORE PICTURES. Take some closeups of the stems, buds, branches, trunk, etc. This will allow a positive ID to be made.
With no other information at this point, I will say emphatically that I do NOT believe this is a Beech tree (Fagus sp.) of any sort. It looks much more like a Hornbeam, and probably one of the European or Asian Hornbeams (Carpinus sp.), due to the very obvious catkins showing out toward the branch tips. This species is a well known heavy suckering tree - historically used as a plant that responds well to coppicing to generate very long straight stems and/or regular sources of firewood due to this behavior. I have one of these here at the Valley - planted about 30 years ago. I have repeatedly cut it to the ground, and it repeatedly resprouts many healthy vigorous ascendent stems. I predict that your plant will do the same, especially if you remove a significant amount of wood at any one time.
Knowing where this plant is growing, and more detailed images to positively ID it, will lead to some quality discussion about what you might want to do with it.