For control of complex networks, controllability criteria are useful to get information about constraints imposed by the network topology. In this work, we study controllability properties for first order dynamics over an undirected random tree network structure. We present a controllability index for the minimum set of driver nodes. We show a criterion for the selection of driver nodes. Then, we demonstrate that regardless the network is growing in time, its structure has a constant controllability index. Finally, an application of this criterion is used for voltage control of active power distribution networks with radial structure.