  Universality is a cornerstone of theories of critical phenomena. It is well
understood in most systems especially in the thermodynamic limit. Finite-size
systems present additional challenges. Even in low dimensions, universality of
the edge and corner contributions to free energies and response functions is
less well understood. The question arises of how universality is maintained in
correction-to-scaling in systems of the same universality class but with very
different corner geometries. 2D geometries deliver the simplest such examples
that can be constructed with and without corners. To investigate how the
presence and absence of corners manifest universality, we analyze the spanning
tree generating function on two finite systems, namely the cobweb and fan
networks. We address how universality can be delivered given that the
finite-size cobweb has no corners while the fan has four. To answer, we appeal
to the Ivashkevich-Izmailian-Hu approach which unifies the generating functions
of distinct networks in terms of a single partition function with twisted
boundary conditions. This unified approach shows that the contributions to the
individual corner free energies of the fan network sum to zero so that it
precisely matches that of the web. Correspondence in each case with results
established by alternative means for both networks verifies the soundness of
the algorithm. Its range of usefulness is demonstrated by its application to
hitherto unsolved problems-namely the exact asymptotic expansions of the
logarithms of the generating functions and the conformal partition functions
for fan and cobweb geometries. Thus, the resolution of a universality puzzle
demonstrates the power of the algorithm and opens up new applications in the
future.
