Let T= (V, E) be a leafless, locally finite rooted directed tree. We associate with T a one parameter family of Dirichlet spaces Hq(q⩾1), which turn out to be Hilbert spaces of vector-valued holomorphic functions defined on the unit disc D in the complex plane. These spaces can be realized as reproducing kernel Hilbert spaces associated with the positive definite kernel (Formula Presented.) where V≺ denotes the set of branching vertices of T, nv denotes the depth of v∈ V in T, and P⟨eroot⟩, Pv(v∈V≺) are certain orthogonal projections. Further, we discuss the question of unitary equivalence of operators Mz(1) and Mz(2) of multiplication by z on Dirichlet spaces Hq associated with directed trees T1 and T2 respectively. © 2017, Springer International Publishing AG.