The central finding is a split between very small and nearly complete subtree clues. The analysis studies pairs of trees through pooled multisets of their k-leaf subtrees, focusing on k = 3 and k = n - 1. With the three-leaf bucket, distinct labelled pairs can share exactly the same information for every n > 4. With the n - 1 bucket, labelled pairs are recoverable when n > 6, while unlabelled pairs are recoverable in the stated classes n > 8, n < 4 and n = 5.
That contrast is about unique recovery, meaning whether one original pair is singled out by the bucket. It is not a result from a biological sample: the mathematical units are tree pairs and their pooled subtree multisets, and no new data were created or analyzed. The findings therefore describe what can be recovered in this specified reconstruction problem. The analysis reports no frequency or probability for ambiguous cases, so it does not say how common either outcome would be in biology.
The stronger clues have a small-case caveat
For unlabelled pairs, the n - 1 bucket does not guarantee recovery at every small size. No pair with four leaves is recoverable. At six, seven and eight leaves, recovery fails only in the pathological cases explicitly identified by the analysis. Outside those exceptions, the reported result covers n > 8, n < 4 and n = 5.
The labelled boundary is different. Up to relabelling, a computer-assisted search found two pair-pair exceptions for five leaves and a single pair-pair exception for six leaves. For labelled pairs with more than six leaves, the n - 1 bucket is reported to recover the pair. The small-case search gives a finite list of exceptions, not a rate at which ambiguous pairs occur.
What the three-leaf bucket keeps
The three-leaf result is not simply a dead end. Labelled pairs are recoverable from the 3-bucket up to a finite sequence of subtree swaps. Those swaps give the ambiguity a specific form: the bucket can remain unchanged after a defined rearrangement of subtrees. Because distinct labelled pairs with the same 3-bucket exist for every n > 4, larger trees do not remove this problem in general.
Even when the full pair is not uniquely recovered, some structure remains visible. For labelled pairs, the 3-bucket recovers the multisets of clusters and cherries. In ordinary terms, these are the leaf groups and paired-leaf features tracked across the two trees. That is partial structural recovery, not necessarily a unique assignment of every shared cluster to one input tree.
A mathematical result with defined boundaries
The reconstruction work uses induction on the number of leaves. Its basic objects are pooled multisets of k-leaf subtrees, and its recovery claims distinguish cases where another pair could share that bucket. The five- and six-leaf labelled results come from a computer-assisted search, while the broader recovery statements are theorem-based.
The n - 1 bucket also carries information about selected features of the trees. If the two input trees differ in height, measured in levels, by at least three, both are recoverable from that bucket. For n > 4, the set of the two root balances, the way their leaves split at the root, is recoverable; for n > 13, the set of pendant depths, the depths of their terminal leaves, is recoverable as well. These results add structural detail to the main recovery theorems, but they do not turn the three-leaf bucket into a unique identifier.
Publication note
The document identifies itself as arXiv version 1 dated 26 Aug 2026. It reports scholarship support for SB and Australian Research Council Discovery Projects support for MH. The manuscript also states that no new data were created or analyzed.
Paper data and sources
Original title: Tree Buckets and the Reconstruction of Pairs of Phylogenetic Trees
Authors: Sky Basire, Michael Hendriksen
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text