  We consider a particular example of interplay between statistical models
related to CFT on one hand, and to the spectral properties of ODE, known as
ODE/IS correspondence, on the other hand. We focus at the representation of
wave functions of Schr\"odinger operators in terms of spectral properties of
associated transfer matrices on "super trees" (the trees whose vertex degree
changes with the distance from the root point). Such trees with varying
branchings encode the structure of the Fock space of the model. We discuss
basic spectral properties of "averaged random matrix ensembles" in terms of
Hermite polynomials for the transfer matrix of super trees. At small "branching
velocities" we have related the problem of paths counting on super trees to the
statistics of area-weighted one-dimensional Dyck paths. We also discuss the
connection of the spectral statistics of random walks on super trees with the
Kardar-Parisi-Zhang scaling.
