  Tensor models are natural generalizations of matrix models. The interactions
and observables in the case of unitary invariant models are generalizations of
matrix traces. Some notable interactions in the literature include the melonic
ones, the tetrahedral one as well as the planar ones in rank three, or
necklaces in even ranks. Here we introduce generalized melonic interactions
which generalize the melonic and necklace interactions. We characterize them as
tree-like gluings of quartic interactions. We also completely characterize the
Feynman graphs which contribute to the large $N$ limit. For a subclass of
generalized melonic interactions called totally unbalanced interactions, we
prove that the large $N$ limit is Gaussian and therefore the Feynman graphs are
in bijection with trees. This result further extends the class of tensor models
which fall into the Gaussian universality class. Another key aspect of tensor
models with generalized melonic interactions is that they can be written as
matrix models without increasing the number of degrees of freedom of the
original tensor models. In the case of totally unbalanced interactions, this
new matrix model formulation in fact decreases the number of degrees of
freedom, meaning that some of the original degrees of freedom are effectively
integrated. We then show how the large $N$ Gaussian behavior can be reproduced
using a saddle point analysis on those matrix models.
