This is exercise II.9.16 from Aluffi's Algebra: Chapter 0.
Before tackling this theorem, I have proved (rather loosely because I don't know much about graph theory) that the Cayley diagram of a free group is free, and that groups acts freely on a tree iff it is a free group. According to the book, with these in mind, we are ready to prove the theorem.
The very first idea that I thought of is to prove that subgroups of a free group act freely on a tree. How can this be done with minimum use of knowledge of graph theory?
Any hints, solutions or reference of other sources is appreciated.