  We discuss de Branges-Rovnyak spaces $\mathcal H(b)$ generated by nonextreme
and rational functions $b$ and local Dirichlet spaces of order $m$ introduced
in [6]. In [6] the authors characterized nonextreme $b$ for which the operator
$Y=S|_{\mathcal H(b)}$, the restriction of the shift operator $S$ on $H^2$ to
$\mathcal H(b)$, is a strict $2m$-isometry and proved that such spaces
$\mathcal H (b)$ are equal to local Dirichlet spaces of order $m$. Here we give
a characterization of local Dirichlet spaces of order $m$ in terms of the
$m$-th derivatives that is a generalization of a known result on local
Dirichlet spaces. We also find explicit formulas for $b$ in the case when
$\mathcal H(b)$ coincides with local Dirichlet space of order $m$ with equality
of norms. Finally, we prove a property of wandering vectors of $Y$ analogous to
the property of wandering vectors of the restriction of $S$ to harmonically
weighted Dirichlet spaces obtained by D. Sarason in [11].
