How to thank a good teacher?

Say if you were a graduate of ABC university, and When you want to thank a teacher of this university because she used to help you a lot in some courses or because she has the same hobbies and interests as yours, and you want to be friends with her:

1. Should you just send a thank you letter to her or buy her a present?


2. If you choose to buy her a present, should it be a brand-named and very expensive one?