Intermediate-size buckets for unlabelled trees

Determine whether there exists an integer k with 3<k<n-1 for which recoverability of pairs of unlabelled phylogenetic trees from their k-buckets is sometimes possible while still admitting an infinite family of counterexamples, and determine the threshold at which this transition occurs.

Background

The paper studies recoverability of pairs of rooted binary phylogenetic trees from pooled multisets of k-leaf subtrees. For unlabelled trees, recovery is impossible from 3-buckets because every 3-leaf unlabelled tree has the same shape, whereas recovery from (n-1)-buckets is possible apart from finitely many small counterexamples.

The authors leave unresolved whether an intermediate bucket size can exhibit a mixed regime in which some pairs are recoverable but infinitely many pairs remain unrecoverable, and where the transition between these behaviours occurs.

References

Similarly, in the unlabelled case, we saw that recoverability is completely impossible from the $3$-bucket, but outside this phenomenon, is there $3<k<n-1$ for which recoverability is sometimes possible, but with an infinite family of counterexamples? If so, at what point does this transition occur?

— Tree Buckets and the Reconstruction of Pairs of Phylogenetic Trees  (2608.25446 - Basire et al., 26 Aug 2026) in Section 5, Discussion