Weight-preserving correspondence between growth orders and forest sums

Construct a weight-preserving correspondence between the growth orders of a fixed rooted forest and the arborescence or forest-sum expansion of the corresponding cofactor, accounting for the order-dependent denominators arising from changing bases.

Background

The order-mass identity equates a sum over growth orders, whose factors depend on a sequence of expanding bases, with a cofactor that can itself be expanded as a sum over arborescences. The paper explicitly observes that no weight-preserving correspondence between these two indexing sets is known.

A successful correspondence would provide a bijective explanation of why the growth-order terms combine into the forest sums. The principal obstacle identified by the paper is that the denominators depend on the entire discovery order and therefore reflect changing domains.

References

Second, a bijective proof of \cref{thm:cofactor} could explain how the growth orders combine into the forest sums of \cref{rem:forest}. The denominators $\prod_\ell\det L[B_{\ell-1}]$ depend on the entire order, so a weight-preserving correspondence would have to account for the changing domains. Connections with loop erasure, such as those in , may be useful for this question.

— The Order Mass Identity of Rooted Forests, with a Direct Proof of the Extended Aldous--Broder Theorem  (2610.13063 - Zheng, 9 Oct 2026) in Section Discussion; related discussion in Remark 4.4 ("Forest sums")