can't prove proposition about transitive trees.

I need to prove: (T,S) is a transitive tree iff (T, S<) is a tree (where S< represents the immediate predecessor relation given by s S< t iff s S t and moreover s S v and v S t for holds for no v in T).

I proved the ==> direction by its contrapositive, but I'm thinking the <== direction is false. What happens if S< simply equals S. We know S< isn't transitive from the definition - so (T,S) wouldn't be a transitive tree. If this isn't a counterexample, let me know what I'm missing here.