  One of the most challenging problems in polymer physics is providing a
theoretical description for the behaviour of rings in dense solutions and
melts. Although it is nowadays well established that the overall size of a ring
in these conditions scales like that of a collapsed globule, there is
compelling evidence that rings may exhibit ramified and tree-like
conformations. In this work I show how to characterise these local tree-like
structures by measuring the local writhing of the rings' segments and by
identifying the patterns of intra-chain contacts. These quantities reveal two
major topological structures: loops and terminal branches which strongly
suggest that the strictly double-folded "lattice animal" picture for rings in
the melt may be replaced by a more relaxed tree-like structure accommodating
loops. In particular, I show that one can identify hierarchically looped
structures whose degree increases linearly with the size of a ring, and that
terminal branches are found to store about 30% of the whole ring mass,
irrespectively of its length. Finally, I draw an analogy between rings in the
melt and slip-linked chains, where contact points are enforced by mobile
slip-links and for which a field-theoretic treatment can be employed to get
some insight into their typical conformations. These findings are ultimately
discussed in the light of recent works on the static structure of rings and on
the existence of inter-ring threadings.
