Problem with "tree" definitions

Solution 1:

In the context of finite graphs I’d call (vii) a tree and (iv) a forest. In a general set-theoretic context I’d apply these terms to (vi) and (iii), respectively. I’ve not seen the term tree used when the partial order isn’t well-founded, but I might call (v) a tree-like poset, (ii) a forest-like poset, and (i) a rootless forest-like poset. More likely though, if I were actually working with such objects, I’d say outright that I was using tree to refer to (v), forest to refer to (ii), and rootless forest to refer to (i). I could then distinguish the more usual objects as well-founded trees/forests, if I had to distinguish them.