Corollary 4

A nonempty graph G is a tree if and only if the groupoid $\varpi_G$ is simply transitive.

Let $a$ be a point of G. By Corollary 1, the following properties are equivalent:

(i) The isotropy group of $\varpi_G$ at $a$ is reduced to the identity element;

(ii) The only loop in G with origin $a$ which is without backtracking is the constant loop with origin $a$.

The groupoid $\varpi_G$ is simply transitive if and only if it is transitive and has property (i) for every $a$. The graph G is a tree if and only if it is connected and has property (ii) for every point $a$. Since the orbits of $\varpi_G$ are identified with the connected components of G, the corollary follows.