In this paper we study the controllability problem for a system that models the vibrations of a controlled tree-shaped network of vibrating elastic strings. The control acts through one of the exterior nodes of the network. With the help of the d'Alembert representation formula for the solutions of the one-dimensional wave equation, we find certain linear relations between the traces of the solutions at the nodes of the network. These relations allow us to prove a weighted observability inequality with weights that may be explicitly computed in terms of the eigenvalues of the associated elliptic problem. We characterize the class of trees for which all those weights are different from zero, which leads to the spectral controllability of the system. Additionally, we consider the same one-node control problem for several networks that are controlled simultaneously.