  We study superstable groups acting on trees. We prove that an action of an
$\omega$-stable group on a simplicial tree is trivial. This shows that an
HNN-extension or a nontrivial free product with amalgamation is not
$\omega$-stable. It is also shown that if $G$ is a superstable group acting
nontrivially on a $\Lambda$-tree, where $\Lambda=\mathbb Z$ or $\Lambda=\mathbb
R$, and if $G$ is either $\alpha$-connected and $\Lambda=\mathbb Z$, or if the
action is irreducible, then $G$ interprets a simple group having a nontrivial
action on a $\Lambda$-tree. In particular if $G$ is superstable and splits as
$G=G_1*_AG_2$, with the index of $A$ in $G_1$ different from 2, then $G$
interprets a simple superstable non $\omega$-stable group.
  We will deal with "minimal" superstable groups of finite Lascar rank acting
nontrivially on $\Lambda$-trees, where $\Lambda=\mathbb Z$ or $\Lambda=\mathbb
R$. We show that such groups $G$ have definable subgroups $H_1 \lhd H_2 \lhd
G$, $H_2$ is of finite index in $G$, such that if $H_1$ is not
nilpotent-by-finite then any action of $H_1$ on a $\Lambda$-tree is trivial,
and $H_2/H_1$ is either soluble or simple and acts nontrivially on a
$\Lambda$-tree. We are interested particularly in the case where $H_2/H_1$ is
simple and we show that $H_2/H_1$ has some properties similar to those of bad
groups.
