  Stability of the branching structure of an RNA molecule is an important
condition for its function. In this letter we show that the melting
thermodynamics of RNA molecules is very sensitive to their branching geometry
for the case of a molecule whose groundstate has the branching geometry of a
Cayley Tree and whose pairing interactions are described by the Go model.
Whereas RNA molecules with a linear geometry melt via a conventional continuous
phase transition with classical exponents, molecules with a Cayley Tree
geometry are found to have a free energy that seems smooth, at least within our
precision. Yet, we show analytically that this free energy in fact has a
mathematical singularity at the stability limit of the ordered structure. The
correlation length appears to diverge on the high-temperature side of this
singularity.
