  Let $\mathscr T=(V, \mathcal E)$ be a leafless, locally finite rooted
directed tree. We associate with $\mathscr T$ a one parameter family of
Dirichlet spaces $\mathscr H_q~(q \geqslant 1)$, which turn out to be Hilbert
spaces of vector-valued holomorphic functions defined on the unit disc $\mathbb
D$ in the complex plane. These spaces can be realized as reproducing kernel
Hilbert spaces associated with the positive definite kernel \begin{eqnarray*}
\kappa_{\mathscr H_q}(z, w) = \sum_{n=0}^{\infty}\frac{(1)_n}{(q)_n}\,{z^n
\overline{w}^n} ~P_{\langle e_{\mathsf{root}}\rangle} + \sum_{v \in V_{\prec}}
\sum_{n=0}^{\infty} \frac{(n_v +2)_n}{(n_v + q+1)_n}\, {z^n
\overline{w}^n}~P_{v}~(z, w \in \mathbb D), \end{eqnarray*} where $V_{\prec}$
denotes the set of branching vertices of $\mathscr T$, $n_v$ denotes the depth
of $v \in V$ in $\mathscr T,$ and $P_{\langle e_{\mathsf{root}}\rangle}$,
$~P_{v}~(v \in V_{\prec})$ are certain orthogonal projections. We also discuss
some structural properties of the operator $\mathscr M_{z, q}$ of
multiplication by $z$ on $\mathscr H_q.$ Further, we discuss the question of
unitary equivalence of operators $\mathscr M^{(1)}_z$ and $\mathscr M^{(2)}_z$
of multiplication by $z$ on Dirichlet spaces $\mathscr H_q$ associated with
directed trees $\mathscr T_1$ and $\mathscr T_2$ respectively.
