The article tackles the problem of inferring information about the unknown structure of a network of dynamic systems under the assumption that the systems are connected according to a tree topology. In particular, the article introduces methodologies to address the presence of hidden (unmeasured) nodes in a scenario where only non-invasive observations are available. By non-invasive observations, it is meant that no known input signal is actively injected into the network. The whole system instead is assumed to be forced by unknown external excitations modeled as stochastic processes. No a priori assumption is made about the number and location of the hidden nodes. Current approaches are capable of consistently inferring the network structure from data, when the dynamics are linear and/or the measurements have a jointly Gaussian distribution. This work provides an approach that can also be applied to networks with nonlinear dynamics and non-Gaussian disturbances. Sufficient conditions are found under which a consistent reconstruction of the topology can be obtained.