Actually, the bark doesn't look like how I remember mulberry. (I used to live in Oklahoma.) But the leaf and the yellow wood do. Farmer Steve's pic is mulberry as I remember it.
Just looked and found there is fruitless mulberry (which would rule out the berry drop) and several varieties which could explain the different bark.
What you cut reminds me of something I used to find in Okla., but can't recall what. I lean toward mulberry on the leaf and yellow wood.