Distances between Extension Spaces of Phylogenetic Trees

Amy D Willis; Maria Alejandra Valdez Cabrera

Distances between Extension Spaces of Phylogenetic Trees

T0 review · 2 major / 2 minor · reviewed 2026-05-23 · grok-4.3

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Pith's one-line read Phylogenetic trees with non-identical leaf sets can be compared by measuring the shortest BHV distance between their extension spaces.

desk verdict The paper defines a distance for trees with non-identical leaves as the shortest BHV distance between extension spaces and supplies a reduced gradient algorithm to compute it, but the algorithm has no established guarantee of reaching the global minimum. read the letter →

arxiv 2407.00227 v1 submitted 2024-06-28 q-bio.QM

Maria Alejandra Valdez Cabrera , Amy D Willis This is my paper

classification q-bio.QM

keywords phylogenetictreesBHVmetricextensionspacestreedistancereducedgradientalgorithmgeneevolutionarydivergence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Phylogenetic trees often differ in their leaf sets because genes are not shared across all species. The BHV metric space already supports analysis of trees that share all leaves, but extension spaces are needed to place trees with missing leaves into the same space. The paper defines the distance between any two such trees as the shortest BHV distance connecting a point in one extension space to a point in the other. It supplies a reduced gradient algorithm that finds this shortest distance in practice. The resulting measure is then used to compare gene trees drawn from organisms across multiple domains of life.

What carries the argument

BHV extension spaces, which place phylogenetic trees with differing leaf sets into one common BHV metric space so that shortest-path distances between them become well-defined.

What would settle it

Apply the reduced gradient algorithm to a collection of gene trees whose leaf sets are known to differ and check whether the returned distances remain stable under small changes in the input trees or under repeated random initializations of the optimizer.

Watch

Extended reading notes

Core claim

We define the distance between two phylogenetic trees with non-identical leaf sets as the shortest BHV distance between their extension spaces, and develop a reduced gradient algorithm to compute this distance. We study the scalability of our algorithm and apply it to analyze gene trees spanning multiple domains of life. Our distance and algorithm offer a fully general, interpretable approach to analyzing both ancient and recent evolutionary divergence.

Load-bearing premise

Extension spaces for trees with non-identical leaf sets can be defined inside the BHV metric space such that the shortest distance between any two spaces is both meaningful and reachable by the reduced gradient algorithm.

Editorial extensions

If this is right

Collections of gene trees that previously could not be placed in one metric space can now be analyzed together.
The algorithm provides a concrete numerical value for evolutionary divergence between genes that do not share all species.
Scalability tests show the method can be run on real multi-domain datasets without requiring all trees to have identical leaves.
The distance remains interpretable because it is expressed directly in the geometry of the BHV space.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

The same shortest-path idea could be tested on other tree metrics once their extension spaces are defined.
Repeated application to many gene trees might expose systematic differences between recent and ancient divergence patterns.
The approach supplies a quantitative handle on the problem of comparing species trees to gene trees when leaf sets are incomplete.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Desk Editor's Note

read the letter

The key point is that this work defines a distance for phylogenetic trees that have different sets of leaves by finding the shortest path in the BHV space between their extension spaces, and they supply a reduced gradient method to calculate it. They also show how it scales and use it on gene trees from multiple domains of life. This is genuinely new because previous BHV tools required the same leaves, and extension spaces were defined but not equipped with a practical distance. The application to real data is a plus, as it shows the method can handle actual biological questions about ancient and recent divergences. The main concern is with the algorithm. The extension spaces are not convex sets, being unions of orthants, so the distance function isn't convex either. First-order methods like reduced gradient can converge to local minima, and there's no proof or argument in the paper that this procedure reaches the global infimum or crosses orthant boundaries correctly. That makes the computed distances potentially unreliable without further checks. This paper is aimed at computational biologists and phylogeneticists who need to compare trees across genes with partial overlap. A reader interested in tree metrics would get value from the definition, but anyone using the numbers should verify the optimization. It deserves peer review because the problem is important and the approach is reasonable, though the optimization part will need scrutiny. I would recommend sending it for review with attention to the convergence properties.

Referee Report

2 major / 2 minor

Summary. The manuscript defines the distance between two phylogenetic trees with non-identical leaf sets as the infimum of BHV distances between points in their respective extension spaces. It introduces a reduced gradient algorithm to compute this distance, reports on the algorithm's scalability, and applies the method to gene trees spanning multiple domains of life.

Significance. If the algorithm is shown to compute the global infimum, the distance provides a principled, fully general extension of BHV-based comparisons to the common case of non-shared leaves; the real-data application demonstrates interpretability for ancient and recent divergences.

major comments (2)

[§3] §3 (Reduced gradient algorithm): the procedure is presented as computing the defined distance (inf{d(x,y) | x ∈ Ext(T1), y ∈ Ext(T2)}), yet no argument, critical-point characterization, or escape-from-boundary analysis is supplied showing that first-order steps on the non-convex union-of-orthants sets reach the global infimum rather than a local minimum.
[§4] §4 (scalability experiments): the reported timings and success rates are obtained on instances whose extension spaces may be convex or have unique minima; no test cases are included that deliberately place multiple orthant boundaries between the trees, so the experiments do not probe whether the algorithm recovers the global distance when local minima exist.

minor comments (2)

[§2] Notation for extension spaces (Ext(T)) is introduced without an explicit statement of the ambient BHV coordinates used when leaves are missing.
Figure captions for the gene-tree application would benefit from explicit mention of the number of missing leaves per pair.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We address each major comment point by point below.

read point-by-point responses

Referee: [§3] §3 (Reduced gradient algorithm): the procedure is presented as computing the defined distance (inf{d(x,y) | x ∈ Ext(T1), y ∈ Ext(T2)}), yet no argument, critical-point characterization, or escape-from-boundary analysis is supplied showing that first-order steps on the non-convex union-of-orthants sets reach the global infimum rather than a local minimum.

Authors: We agree that the manuscript does not supply a formal argument, critical-point characterization, or escape-from-boundary analysis establishing that the reduced gradient algorithm reaches the global infimum on the non-convex union-of-orthants geometry. The optimization problem is non-convex, and the current presentation treats the algorithm as a practical method for computing the distance without a global optimality guarantee. In the revised version we will update §3 to state explicitly that the procedure yields a local minimum (hence an upper bound on the defined infimum) and to discuss the implications of possible local minima for the distance definition. revision: partial
Referee: [§4] §4 (scalability experiments): the reported timings and success rates are obtained on instances whose extension spaces may be convex or have unique minima; no test cases are included that deliberately place multiple orthant boundaries between the trees, so the experiments do not probe whether the algorithm recovers the global distance when local minima exist.

Authors: We accept this observation. The experiments in §4 measure runtime and success rates across varying leaf-set overlaps and tree sizes but do not construct instances with multiple orthant boundaries chosen to induce local minima. We will revise §4 to incorporate a new set of test cases deliberately designed with multiple orthant boundaries and will report the algorithm's performance on those instances, thereby providing empirical evidence on behavior when local minima may exist. revision: yes

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; new distance defined directly in BHV space with independent algorithm

full rationale

The paper introduces a definition of distance between trees with non-identical leaf sets as the infimum of BHV distances between their extension spaces, followed by a reduced gradient algorithm for computation. This is a direct definitional step with no reduction to fitted inputs, self-citations, or prior ansatzes by the same authors. The BHV space is an external reference, and the extension spaces are constructed within it without the result being equivalent to the inputs by construction. No load-bearing self-citation chains or self-definitional loops appear.

Assumptions & free parameters 0 free parameters · 0 assumptions · 0 invented entities

Based on abstract only; no specific free parameters, axioms, or invented entities can be identified without the full text. Standard phylogenetic assumptions (e.g., tree structures) are implicit but not detailed.

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Cite this review

Pith. "Pith review of Distances between Extension Spaces of Phylogenetic Trees." pith.science (2026). https://pith.science/paper/2407.00227

@misc{pith2026240700227,
  author       = {Pith},
  title        = {Pith review of: Distances between Extension Spaces of Phylogenetic Trees},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/2407.00227}},
  note         = {Machine review of arXiv:2407.00227}
}

read the original abstract

Phylogenetic trees summarize evolutionary relationships between organisms, and tools to analyze collections of phylogenetic trees enable contrasts between different genes' ancestry. The BHV metric space has enabled the analysis of collections of trees that share a common set of leaves, but many genes are not shared, even between closely related species. BHV extension spaces represent trees with non-identical leaf sets in a common BHV space, but limited analytical tools exist for extension spaces. We define the distance between two phylogenetic trees with non-identical leaf sets as the shortest BHV distance between their extension spaces, and develop a reduced gradient algorithm to compute this distance. We study the scalability of our algorithm and apply it to analyze gene trees spanning multiple domains of life. Our distance and algorithm offer a fully general, interpretable approach to analyzing both ancient and recent evolutionary divergence.

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Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We define the distance between two phylogenetic trees with non-identical leaf sets as the shortest BHV distance between their extension spaces, and develop a reduced gradient algorithm to compute this distance.
IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the extension space of T in the BHV space T_N is E_N_T = {T' in T_N | d_T'(ℓ1,ℓ2)=d_T(ℓ1,ℓ2) for all ℓ1,ℓ2 in L}

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matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

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Reviewed May 23, 2026 · model on record in the stance chip above.