Let $\cal G$ be a pseudogroup defined on a tree $Z$, and let
$\Gamma$ be a finite set of generators for $\cal G$.
The reduced fundamental group $\pibar(\Gamma)$
of $\Gamma$ is defined here.
I give
a new and experimentally inspired
proof of a result of Levitt: If $\pibar(\Gamma)$ is a
free group,
there exists a finite set of generators $\Psi$ for $\cal G$ such that
$\pibar(\Psi)$ is free on the set $\Psi$. If $\Psi$ has no dead ends, it is an
interval exchange.
¶ Like Gaboriau, Levitt and Paulin [Gaboriau et al. 1992], I prove
that if $G$ is a finitely presented group acting freely on an $\R$-tree and
$\Gamma$ is a corresponding set of pseudogroup generators, we're in one of the
following situations:
either $G$
splits as a free product with a noncyclic free abelian summand, or $\Gamma$
can be reduced to an interval exchange by normalizing and removing a finite
number of dead ends, or the process of removing dead ends from $\Gamma$
does not terminate in a finite number of steps.