  An odd Coxeter group $W$ is one which admits a Coxeter system $(W,S)$ for
which all the exponents $m_{ij}$ are either odd or infinity. The paper
investigates the family of odd Coxeter groups whose associated labeled graphs
$\mathcal{V}_{(W,S)}$ are trees. It is known that two Coxeter groups in this
family are isomorphic if and only if they admit Coxeter systems having the same
rank and the same multiset of finite exponents. In particular, each group in
this family is isomorphic to a group that admits a Coxeter system whose
associated labeled graph is a star shaped tree. We give the complete
description of the automorphism group of this group, and derive a sufficient
condition for the splitting of the automorphism group as a semi-direct product
of the inner and the outer automorphism groups. As applications, we prove that
Coxeter groups in this family satisfy the $R_\infty$-property and are
(co)-Hopfian. We compare structural properties, automorphism groups,
$\R_\infty$-property and (co)-Hopfianity of a special odd Coxeter group whose
only finite exponent is three with the braid group and the twin group.
