What kind of nut or Fruit are these?
I'll tell you what they look like to me, but I'm not very confident in my answers. The first one looks like it could be an oak gall or some sort of a gall on a tree. Was it hollow and light? Specifically it looks a lot like the oak apple gall.
The second one looks like a hickory or even a pecan still in the green husk.
I would recommend that you contact Dr. Barron Rector who is a range specialist. He's excellent at plant identification and would know if deer would like to eat whatever these things turn out to be. His email is firstname.lastname@example.org and his number is 979.845.2755.