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This paper studies the Dirichlet-type space and the generalized
Dirichlet-type space whose defect operators are of finite ranks.

The Dirichlet-type space $D(\mu)$ is defined by a nonnegative Borel measure
$\mu$ on the closed unit disc. When $\mu$ is a finitely atomic measure, we
characterize the conditions for the de Branges-Rovnyak space $\mathcal{H}(B)$
and the $D(\mu)$ space to be the same.

The generalized Dirichlet-type space $\mathcal{D}_{\vec{\mu}}$ is determined
by a tuple $\vec{\mu}$ of distributions on the unit circle. If $\vec{\mu} =
(\frac{|dz|}{2\pi}, \mu_1, \ldots, \mu_{n-1})$ is an allowable $n$-tuple such
that $(M_z, \mathcal{D}_{\vec{\mu}})$ is expansive and the defect operator of
$(M_z, \mathcal{D}_{\vec{\mu}})$ has finite rank, then we show that $n$ is
even, $\mu_{n-1}$ is a positive measure, the distributions $\mu_i$ $(i = 1,
\ldots, n-1)$ are supported on a finite set contained in the unit circle, and
the order of each $\mu_i$ doesn't exceed $\frac{n}{2}-1$. We show that the
converse of this is true when $n=2$, but when $n \geq 3$, the converse is not
true. To achieve this, we use the characterization of the higher order
isometric $(M_z,\mathcal{H}(B))$ that are finite rank perturbations of
isometric operators obtained in \cite{LGR}. Our investigation also provides
many interesting examples of normalized allowable $2m$-tuples $\vec{\mu}$.