Obviously the OP uses book Modal logic By Patrick Blackburn, Maarten de Rijke, Yde Venema
- it is exercise 1.1.4 from this book.
I think your counterexample works, but if you change the claim to:
"Let S be a SPO on T. (T,S) is a transitive tree if (T,S<) is a tree."
then it should work.