Trees and the Nielsen-Schreier Theorem

Abstract

A graph Γ consists of two sets A (directed edges) and V (vertices) together with two functions

$$A \to A,\alpha \to \alpha $$

$$A \to V \times V,\alpha \to (i\left( \alpha \right),t\left( \alpha \right))$$

which satisfy $\bar \bar a = \alpha ,\bar \alpha \ne \alpha$ for each α in A. The vertices i(α), t(α) are the initial and terminal vertices of the directed edge α, and $\bar \alpha$ is the reverse of α. Henceforth we refer to directed edges simply as edges.