  For unitary operators $U_0,U$ in Hilbert spaces ${\mathcal H}_0,{\mathcal H}$
and identification operator $J:{\mathcal H}_0\to{\mathcal H}$, we present
results on the derivation of a Mourre estimate for $U$ starting from a Mourre
estimate for $U_0$ and on the existence and completeness of the wave operators
for the triple $(U,U_0,J)$. As an application, we determine spectral and
scattering properties of a class of anisotropic quantum walks on homogenous
trees of odd degree with evolution operator $U$. In particular, we establish a
Mourre estimate for $U$, obtain a class of locally $U$-smooth operators, and
prove that the spectrum of $U$ covers the whole unit circle and is purely
absolutely continuous, outside possibly a finite set where $U$ may have
eigenvalues of finite multiplicity. We also show that (at least) three
different choices of free evolution operators $U_0$ are possible for the proof
of the existence and completeness of the wave operators.
