islands connected by Josephson junctions and have a tree-like topological structure containing no loops. Enhancements of superconductive gap over specific branches of the networks and sharp increases of pair currents are the main signatures of the coherent states and, in order to unambiguously attribute the observed effects to branches being embedded in the networks, comparisons with geometrically equivalent, but isolated, counterparts are reported. Tuning the Josephson coupling energy by an external magnetic field generates increases of the Josephson currents, along the above mentioned specific branches, which follow a functional dependence typical of phase transitions. Results are presented for double comb and star geometry networks and in both cases the observed effects provide positive quantitative evidence of the predictions of existing theoretical models.