  The unitary equivalence of $2$-isometric operators satisfying the so-called
kernel condition is characterized. It relies on a model for such operators
built on operator valued unilateral weighted shifts and on a characterization
of the unitary equivalence of operator valued unilateral weighted shifts in a
fairly general context. A complete system of unitary invariants for
$2$-isometric weighted shifts on rooted directed trees satisfying the kernel
condition is provided. It is formulated purely in the langauge of graph-theory,
namely in terms of certain generation branching degrees. The membership of the
Cauchy dual operators of $2$-isometries in classes $C_{0 \cdot}$ and $C_{\cdot
0}$ is also studied.
