Properties for the Frechet Mean in Billera-Holmes-Vogtmann Treespace

Aisha Ashfaq; Corrine Yap; Ella Pavlechko; Joyce Chiu; Katherine St. John; Keith Thompson; Mahedi Kaiser; Maria Anaya; Max Shoji Ohsawa; Megan Owen

arxiv: 1907.05937 · v1 · pith:TE6I2IG2new · submitted 2019-07-12 · 🧮 math.CO · q-bio.PE

Properties for the Frechet Mean in Billera-Holmes-Vogtmann Treespace

Maria Anaya , Olga Anipchenko-Ulaj , Aisha Ashfaq , Joyce Chiu , Mahedi Kaiser , Max Shoji Ohsawa , Megan Owen , Ella Pavlechko

show 4 more authors

Katherine St. John Shivam Suleria Keith Thompson Corrine Yap

This is my paper

Pith reviewed 2026-05-24 22:11 UTC · model grok-4.3

classification 🧮 math.CO q-bio.PE

keywords Frechet meanBHV treespaceweighted treesorthant spacesedge conditionsphylogenetic treesgeodesic distance

0 comments

The pith

Necessary and sufficient conditions for an edge to be in the Frechet mean are inequalities on the weights of the edges.

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

The paper derives the first conditions that tell exactly when an edge must appear in the intrinsic average of a set of weighted trees. This average, called the Frechet mean, is the unique point in BHV treespace that minimizes the sum of squared geodesic distances to the given trees. Until now only iterative approximation methods existed, with no way to certify in advance which edges the mean would contain. The new conditions are simple inequalities on edge weights and immediately identify the orthant that must contain the mean, turning an open search into a preprocessing filter. The same style of conditions extends directly to arbitrary orthant spaces.

Core claim

We give the first necessary and sufficient conditions for an edge to be in the Frechet mean. The conditions are in the form of inequalities on the weights of the edges. These conditions provide a pre-processing step for finding the treespace orthant containing the Frechet mean. This work generalizes to orthant spaces.

What carries the argument

The weight inequalities that serve as necessary and sufficient conditions for an edge to belong to the Frechet mean.

If this is right

The orthant containing the Frechet mean can be identified before any iterative search begins.
The conditions supply a polynomial-time filter that narrows the combinatorial search space for the mean.
The same characterization applies verbatim to the Frechet mean in any orthant space.
Once the orthant is known, existing approximation algorithms can be restricted to that orthant.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The inequalities may combine with existing solvers to produce exact rather than approximate means in small instances.
In phylogenetic applications the conditions could reduce the cost of averaging large sets of evolutionary trees.
Analogous weight-based tests may exist for other intrinsic statistics on the same space.

Load-bearing premise

The Frechet mean exists and is unique in BHV treespace.

What would settle it

A collection of trees together with an edge that satisfies the stated weight inequalities yet lies outside the true Frechet mean, or an edge that fails the inequalities yet appears in the mean.

Figures

Figures reproduced from arXiv: 1907.05937 by Aisha Ashfaq, Corrine Yap, Ella Pavlechko, Joyce Chiu, Katherine St. John, Keith Thompson, Mahedi Kaiser, Maria Anaya, Max Shoji Ohsawa, Megan Owen, Olga Anipchenko-Ulaj, Shivam Suleria.

**Figure 2.** Figure 2: a) Tree T with common split Y |Z corresponding to edge e = {u, v}. Deleting edge e splits tree T into b) tree T Y with leaves Y ∪ u and c) tree T Z with leaves Z ∪ v. f(T) = Xr i=1 d(T, Ti) 2 (3) where d is the BHV distance. Sturm [33] showed that the Fr´echet mean is unique on globally non-postivively curved spaces, such as the BHV treespace [6]. Baˇc´ak [2] and Miller et al. [22] independently adapted S… view at source ↗

**Figure 3.** Figure 3: (a) An example of three trees with the same topology on 5 leaves, [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

**Figure 4.** Figure 4: (a) Three trees, T1, T2, and T3, are in three different orthants that share an axis, and one tree, T4, lies on this axis. The translated log maps based at T1, T2, and T3 are shown in (b), (c), and (d), respectively. In each log map, the two orthants not containing the base tree have been identified with each other, and the shared axis is extended in the negative direction. The tangent spaces shown in (b), … view at source ↗

read the original abstract

The Billera-Holmes-Vogtmann (BHV) space of weighted trees can be embedded in Euclidean space, but the extrinsic Euclidean mean often lies outside of treespace. Sturm showed that the intrinsic Frechet mean exists and is unique in treespace. This Frechet mean can be approximated with an iterative algorithm, but bounds on the convergence of the algorithm are not known, and there is no other known polynomial algorithm for computing the Frechet mean nor even the edges present in the mean. We give the first necessary and sufficient conditions for an edge to be in the Frechet mean. The conditions are in the form of inequalities on the weights of the edges. These conditions provide a pre-processing step for finding the treespace orthant containing the Frechet mean. This work generalizes to orthant spaces.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper supplies the first necessary and sufficient weight inequalities that tell which edges belong to the Fréchet mean in BHV treespace.

read the letter

The main takeaway is that the authors derive necessary and sufficient conditions, in the form of inequalities on edge weights, for an edge to appear in the Fréchet mean. Earlier papers handled existence, uniqueness, and iterative approximation but stopped short of this explicit edge-inclusion test. The result gives a preprocessing step that narrows the orthant search for the mean, and the authors note it extends to general orthant spaces. That is the concrete advance here. It rests on Sturm's theorem, which is cited cleanly and is the standard reference for uniqueness in this CAT(0) space. The conditions appear to come directly from the orthant geometry rather than from fitted parameters or circular self-reference. The paper is short on examples or proof sketches in the abstract, so the tightness of the inequalities and any boundary cases are not visible yet. If the full derivation holds up, the preprocessing claim is useful; if gaps exist in the argument, they would be the main thing to fix. This is aimed at computational phylogenetics groups that already work with BHV space and need faster ways to locate the mean. A reader who wants a practical handle on orthant location will get value from it even without a full polynomial algorithm. It is not a complete solver with convergence rates, but the edge criteria are a distinct step. I would bring the paper to a reading group focused on geometric methods in biology. I would not cite it in my own work in the next year because the scope is narrow. The claim is specific enough and the area active enough that it deserves referee time rather than a desk reject.

Referee Report

0 major / 1 minor

Summary. The manuscript derives the first necessary and sufficient conditions, expressed as inequalities involving edge weights, that characterize precisely which edges appear in the (unique) Fréchet mean of a collection of trees in BHV treespace. The argument relies on the orthant geometry of the space and on Sturm's theorem guaranteeing existence and uniqueness of the mean; the conditions are positioned as a preprocessing step that identifies the orthant containing the mean, with a noted generalization to general orthant spaces.

Significance. If the stated conditions are correctly derived and verified, the result supplies an explicit, checkable characterization of the support of the Fréchet mean that was previously unavailable. This would constitute a concrete algorithmic advance over purely iterative approximation methods whose convergence rates remain unknown, and the necessary-and-sufficient form together with the orthant-space generalization strengthens the contribution.

minor comments (1)

[Abstract] The abstract asserts that the conditions are 'necessary and sufficient' and 'in the form of inequalities on the weights of the edges'; the manuscript should include an explicit statement of these inequalities (e.g., as a numbered theorem) together with the proof that they are both necessary and sufficient.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript and for recommending acceptance. The referee's summary correctly identifies the main contribution: the first necessary and sufficient conditions, expressed as inequalities on edge weights, that characterize the edges present in the unique Fréchet mean.

Circularity Check

0 steps flagged

No significant circularity; derivation from orthant geometry definition

full rationale

The paper derives necessary and sufficient weight inequalities characterizing edges in the Fréchet mean directly from the definition of the mean in BHV treespace orthant geometry. It invokes Sturm's theorem (an external result on CAT(0) spaces) only for existence and uniqueness of the target object, with no self-citation load-bearing the central claim. No equations reduce by construction to fitted inputs, renamed patterns, or ansatzes smuggled via prior work by the same authors. The result is self-contained against the stated geometric definition.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on Sturm's existence and uniqueness theorem for the Fréchet mean in treespace and on the piecewise-Euclidean orthant structure of BHV space; no free parameters or new entities are introduced in the abstract.

axioms (1)

domain assumption The Fréchet mean exists and is unique in BHV treespace
Invoked from Sturm to guarantee a well-defined mean whose edges can be characterized.

pith-pipeline@v0.9.0 · 5717 in / 1210 out tokens · 25968 ms · 2026-05-24T22:11:29.478680+00:00 · methodology

discussion (0)

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

?

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

We give the first necessary and sufficient conditions for an edge to be in the Fréchet mean. The conditions are in the form of inequalities on the weights of the edges.
IndisputableMonolith/Foundation/AbsoluteFloorClosure.lean reality_from_one_distinction unclear

?

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

Theorem 4 ([3, Theorem 3]). ... tree μ is the Fréchet mean ... if and only if: (i) ... ⟨wF, ∑ Ψ...⟩ ≤ 0 (ii) Φ(μ;μ) = 1/r ∑ ΦS(T;μ)

What do these tags mean?

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.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages · 2 internal anchors

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Allen and M

B. Allen and M. Steel. Subtree transfer operations and their induced metrics on evolutionary trees. Annals of Combinatorics , 5:1–13, 2001

work page 2001
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Computing medians and means in hadamard spaces

Miroslav Bac´ ak. Computing medians and means in hadamard spaces. SIAM Journal on Optimization, 24(3):1542–1566, 2014

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The logarithm map, its limits and Fr´ echet means in orthant spaces

Dennis Barden and Huiling Le. The logarithm map, its limits and Fr´ echet means in orthant spaces. Proceedings of the London Mathematical Society, 117(4):751–789, 2018

work page 2018
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Central limit theorems for Fr´ echet means in the space of phylogenetic trees

Dennis Barden, Huiling Le, and Megan Owen. Central limit theorems for Fr´ echet means in the space of phylogenetic trees. Electronic Journal of Probability, 18(25):1–25, 2013

work page 2013
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Limiting behaviour of Fr´ echet means in the space of phylogenetic trees

Dennis Barden, Huiling Le, and Megan Owen. Limiting behaviour of Fr´ echet means in the space of phylogenetic trees. Annals of the Institute of Statistical Mathematics , pages 1–31, 2016

work page 2016
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Billera, S.P

L.J. Billera, S.P. Holmes, and K. Vogtmann. Geometry of the space of phylogenetic trees. Advances in Applied Mathematics , 27:733–767, 2001

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On the computational complexity of the rooted subtree prune and regraft distance

Magnus Bordewich and Charles Semple. On the computational complexity of the rooted subtree prune and regraft distance. Annals of Combintorics , 8:409–423, 2004

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Daniel G. Brown and Megan Owen. Mean and variance of phylogenetic trees. arXiv preprint arXiv:1708.00294, 2017

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O. Peter Buneman. The recovery of trees from measures of dissimilarity. Mathematics in the Archaeological and Historical Sciences, pages 387–395, 1971. 26

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Geodesic atlas-based labeling of anatomical trees: Application and evaluation on airways extracted from ct

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SPR distance com- putation for unrooted trees

Glenn Hickey, Frank Dehne, Andrew Rau-Chaplin, and Christian Blouin. SPR distance com- putation for unrooted trees. Evolutionary Bioinformatics, 4:17–27, 2008

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Mat- tingly, Ezra Miller, James Nolen, Megan Owen, Vic Patrangenaru, Sean Skwerer, et al

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Sticky central limit theorems at isolated hyperbolic planar singularities

Stephan Huckemann, Jonathan Mattingly, Ezra Miller, James Nolen, et al. Sticky central limit theorems at isolated hyperbolic planar singularities. Electronic Journal of Probability , 20, 2015

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The shape of phylogenetic treespace

Katherine St John. The shape of phylogenetic treespace. Systematic Biology, 66(1):e83, 2017

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Mapping phylogenetic trees to reveal distinct patterns of evolution

Michelle Kendall and Caroline Colijn. Mapping phylogenetic trees to reveal distinct patterns of evolution. bioRxiv, page 026641, 2015

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A simulation comparison of phylogeny algorithms under equal and unequal evolutionary rates

Mary K Kuhner and Joseph Felsenstein. A simulation comparison of phylogeny algorithms under equal and unequal evolutionary rates. Molecular Biology and Evolution , 11(3):459–468, 1994

work page 1994
[22]

Scott Provan

Ezra Miller, Megan Owen, and J. Scott Provan. Polyhedral computational geometry for aver- aging metric phylogenetic trees. Advances in Applied Mathematics , 68:51–91, 2015

work page 2015
[23]

Tom M.W. Nye. Principal components analysis in the space of phylogenetic trees. The Annals of Statistics, pages 2716–2739, 2011

work page 2011
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Tom M.W. Nye. An algorithm for constructing principal geodesics in phylogenetic treespace. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 11(2):304–315, 2014. 27

work page 2014
[25]

Tom M.W. Nye. Convergence of random walks to Brownian motion in phylogenetic tree-space. arXiv preprint arXiv:1508.02906 , 2015

work page internal anchor Pith review Pith/arXiv arXiv 2015
[26]

Principal component analysis and the locus of the Fr´ echet mean in the space of phylogenetic trees

Tom MW Nye, Xiaoxian Tang, Grady Weyenberg, and Ruriko Yoshida. Principal component analysis and the locus of the Fr´ echet mean in the space of phylogenetic trees. Biometrika, 104(4):901–922, 2017

work page 2017
[27]

Distance Computation in the Space of Phylogenetic Trees

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Computing geodesic distances in tree space

Megan Owen. Computing geodesic distances in tree space. SIAM Journal on Discrete Math- ematics, 25(4):1506–1529, 2011

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Scott Provan

Megan Owen and J. Scott Provan. A fast algorithm for computing geodesic distances in tree space. IEEE/ACM Transactions on Computational Biology & Bioinformatics, 8:2–13, January 2011

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Phylogenetics, volume 24 of Oxford Lecture Series in Mathe- matics and its Applications

Charles Semple and Mike Steel. Phylogenetics, volume 24 of Oxford Lecture Series in Mathe- matics and its Applications . Oxford University Press, Oxford, 2003

work page 2003
[31]

Tree-oriented analysis of brain artery structure

Sean Skwerer, Elizabeth Bullitt, Stephan Huckemann, Ezra Miller, Ipek Oguz, Megan Owen, Vic Patrangenaru, Scott Provan, and James Stephen Marron. Tree-oriented analysis of brain artery structure. Journal of Mathematical Imaging and Vision , 50(1-2):126–143, 2014

work page 2014
[32]

Sean Skwerer, Scott Provan, and J.S. Marron. Relative optimality conditions and algorithms for treespace Fr´ echet means.SIAM Journal on Optimization , 28(2):959–988, 2018

work page 2018
[33]

Probability measures on metric spaces of nonpositive curvature

Karl-Theodor Sturm. Probability measures on metric spaces of nonpositive curvature. In Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002) , volume 338 of Contemp. Math., pages 357–390. Amer. Math. Soc., Providence, RI, 2003

work page 2002
[34]

Geodesics in the space of trees, 2007

Karen Vogtmann. Geodesics in the space of trees, 2007. Available at http://pi.math. cornell.edu/~vogtmann/papers/TreeGeodesicss/geodesics07.pdf. Last accessed July 31, 2018

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[35]

Conﬁdence sets for phylogenetic trees

Amy Willis. Conﬁdence sets for phylogenetic trees. Journal of the American Statistical Asso- ciation, pages 1–10, 2018

work page 2018
[36]

Uncertainty in phylogenetic tree estimates

Amy Willis and Rayna Bell. Uncertainty in phylogenetic tree estimates. Journal of Computa- tional and Graphical Statistics , 27(3):542–552, 2018. 28

work page 2018

[1] [1]

Allen and M

B. Allen and M. Steel. Subtree transfer operations and their induced metrics on evolutionary trees. Annals of Combinatorics , 5:1–13, 2001

work page 2001

[2] [2]

Computing medians and means in hadamard spaces

Miroslav Bac´ ak. Computing medians and means in hadamard spaces. SIAM Journal on Optimization, 24(3):1542–1566, 2014

work page 2014

[3] [3]

The logarithm map, its limits and Fr´ echet means in orthant spaces

Dennis Barden and Huiling Le. The logarithm map, its limits and Fr´ echet means in orthant spaces. Proceedings of the London Mathematical Society, 117(4):751–789, 2018

work page 2018

[4] [4]

Central limit theorems for Fr´ echet means in the space of phylogenetic trees

Dennis Barden, Huiling Le, and Megan Owen. Central limit theorems for Fr´ echet means in the space of phylogenetic trees. Electronic Journal of Probability, 18(25):1–25, 2013

work page 2013

[5] [5]

Limiting behaviour of Fr´ echet means in the space of phylogenetic trees

Dennis Barden, Huiling Le, and Megan Owen. Limiting behaviour of Fr´ echet means in the space of phylogenetic trees. Annals of the Institute of Statistical Mathematics , pages 1–31, 2016

work page 2016

[6] [6]

Billera, S.P

L.J. Billera, S.P. Holmes, and K. Vogtmann. Geometry of the space of phylogenetic trees. Advances in Applied Mathematics , 27:733–767, 2001

work page 2001

[7] [7]

On the computational complexity of the rooted subtree prune and regraft distance

Magnus Bordewich and Charles Semple. On the computational complexity of the rooted subtree prune and regraft distance. Annals of Combintorics , 8:409–423, 2004

work page 2004

[8] [8]

Martin R Bridson and Andr´ e Haeﬂiger.Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenshaften, vol. 319 . Springer, 1999

work page 1999

[9] [9]

Mean and Variance of Phylogenetic Trees

Daniel G. Brown and Megan Owen. Mean and variance of phylogenetic trees. arXiv preprint arXiv:1708.00294, 2017

work page internal anchor Pith review Pith/arXiv arXiv 2017

[10] [10]

Peter Buneman

O. Peter Buneman. The recovery of trees from measures of dissimilarity. Mathematics in the Archaeological and Historical Sciences, pages 387–395, 1971. 26

work page 1971

[11] [11]

On comput- ing the nearest neighbor interchange distance

Bhaskar DasGupta, Xin He, Tao Jiang, Ming Li, John Tromp, and Louxin Zhang. On comput- ing the nearest neighbor interchange distance. In D.Z. Du, P.M. Pardalos, and J. Wang, editors, Proceedings of the DIMACS Workshop on Discrete Problems with Medical Applications , vol- ume 55 of DIMACS Series in Discrete Mathematics and Theoretical Computer Science , pa...

work page 2000

[12] [12]

SageMath, the Sage Mathematics Software System (Version 6.10) , 2015

The Sage Developers. SageMath, the Sage Mathematics Software System (Version 6.10) , 2015. http://www.sagemath.org

work page 2015

[13] [13]

Wille, Laura H

Aasa Feragen, Megan Owen, Jens Petersen, Mathilde M.W. Wille, Laura H. Thomsen, Asger Dirksen, and Marleen de Bruijne. Tree-space statistics and approximations for large-scale analysis of anatomical trees. In International Conference on Information Processing in Medical Imaging, pages 74–85. Springer, 2013

work page 2013

[14] [14]

Geodesic atlas-based labeling of anatomical trees: Application and evaluation on airways extracted from ct

Aasa Feragen, Jens Petersen, Megan Owen, Pechin Lo, Laura Hohw¨ u Thomsen, Mathilde Marie Winkler Wille, Asger Dirksen, and Marleen de Bruijne. Geodesic atlas-based labeling of anatomical trees: Application and evaluation on airways extracted from ct. IEEE transactions on medical imaging, 34(6):1212–1226, 2015

work page 2015

[15] [15]

Les ´ el´ ements al´ eatoires de nature quelconque dans un espace distanci´ e.Ann

Maurice Fr´ echet. Les ´ el´ ements al´ eatoires de nature quelconque dans un espace distanci´ e.Ann. Inst. H. Poincar´ e, 10(3):215–310, 1948

work page 1948

[16] [16]

SPR distance com- putation for unrooted trees

Glenn Hickey, Frank Dehne, Andrew Rau-Chaplin, and Christian Blouin. SPR distance com- putation for unrooted trees. Evolutionary Bioinformatics, 4:17–27, 2008

work page 2008

[17] [17]

Mat- tingly, Ezra Miller, James Nolen, Megan Owen, Vic Patrangenaru, Sean Skwerer, et al

Thomas Hotz, Stephan Huckemann, Huiling Le, James Stephen Marron, Jonathan C. Mat- tingly, Ezra Miller, James Nolen, Megan Owen, Vic Patrangenaru, Sean Skwerer, et al. Sticky central limit theorems on open books. The Annals of Applied Probability , 23(6):2238–2258, 2013

work page 2013

[18] [18]

Sticky central limit theorems at isolated hyperbolic planar singularities

Stephan Huckemann, Jonathan Mattingly, Ezra Miller, James Nolen, et al. Sticky central limit theorems at isolated hyperbolic planar singularities. Electronic Journal of Probability , 20, 2015

work page 2015

[19] [19]

The shape of phylogenetic treespace

Katherine St John. The shape of phylogenetic treespace. Systematic Biology, 66(1):e83, 2017

work page 2017

[20] [20]

Mapping phylogenetic trees to reveal distinct patterns of evolution

Michelle Kendall and Caroline Colijn. Mapping phylogenetic trees to reveal distinct patterns of evolution. bioRxiv, page 026641, 2015

work page 2015

[21] [21]

A simulation comparison of phylogeny algorithms under equal and unequal evolutionary rates

Mary K Kuhner and Joseph Felsenstein. A simulation comparison of phylogeny algorithms under equal and unequal evolutionary rates. Molecular Biology and Evolution , 11(3):459–468, 1994

work page 1994

[22] [22]

Scott Provan

Ezra Miller, Megan Owen, and J. Scott Provan. Polyhedral computational geometry for aver- aging metric phylogenetic trees. Advances in Applied Mathematics , 68:51–91, 2015

work page 2015

[23] [23]

Tom M.W. Nye. Principal components analysis in the space of phylogenetic trees. The Annals of Statistics, pages 2716–2739, 2011

work page 2011

[24] [24]

Tom M.W. Nye. An algorithm for constructing principal geodesics in phylogenetic treespace. IEEE/ACM Transactions on Computational Biology and Bioinformatics, 11(2):304–315, 2014. 27

work page 2014

[25] [25]

Tom M.W. Nye. Convergence of random walks to Brownian motion in phylogenetic tree-space. arXiv preprint arXiv:1508.02906 , 2015

work page internal anchor Pith review Pith/arXiv arXiv 2015

[26] [26]

Principal component analysis and the locus of the Fr´ echet mean in the space of phylogenetic trees

Tom MW Nye, Xiaoxian Tang, Grady Weyenberg, and Ruriko Yoshida. Principal component analysis and the locus of the Fr´ echet mean in the space of phylogenetic trees. Biometrika, 104(4):901–922, 2017

work page 2017

[27] [27]

Distance Computation in the Space of Phylogenetic Trees

Megan Owen. Distance Computation in the Space of Phylogenetic Trees . PhD thesis, Cornell University, 2008

work page 2008

[28] [28]

Computing geodesic distances in tree space

Megan Owen. Computing geodesic distances in tree space. SIAM Journal on Discrete Math- ematics, 25(4):1506–1529, 2011

work page 2011

[29] [29]

Scott Provan

Megan Owen and J. Scott Provan. A fast algorithm for computing geodesic distances in tree space. IEEE/ACM Transactions on Computational Biology & Bioinformatics, 8:2–13, January 2011

work page 2011

[30] [30]

Phylogenetics, volume 24 of Oxford Lecture Series in Mathe- matics and its Applications

Charles Semple and Mike Steel. Phylogenetics, volume 24 of Oxford Lecture Series in Mathe- matics and its Applications . Oxford University Press, Oxford, 2003

work page 2003

[31] [31]

Tree-oriented analysis of brain artery structure

Sean Skwerer, Elizabeth Bullitt, Stephan Huckemann, Ezra Miller, Ipek Oguz, Megan Owen, Vic Patrangenaru, Scott Provan, and James Stephen Marron. Tree-oriented analysis of brain artery structure. Journal of Mathematical Imaging and Vision , 50(1-2):126–143, 2014

work page 2014

[32] [32]

Sean Skwerer, Scott Provan, and J.S. Marron. Relative optimality conditions and algorithms for treespace Fr´ echet means.SIAM Journal on Optimization , 28(2):959–988, 2018

work page 2018

[33] [33]

Probability measures on metric spaces of nonpositive curvature

Karl-Theodor Sturm. Probability measures on metric spaces of nonpositive curvature. In Heat kernels and analysis on manifolds, graphs, and metric spaces (Paris, 2002) , volume 338 of Contemp. Math., pages 357–390. Amer. Math. Soc., Providence, RI, 2003

work page 2002

[34] [34]

Geodesics in the space of trees, 2007

Karen Vogtmann. Geodesics in the space of trees, 2007. Available at http://pi.math. cornell.edu/~vogtmann/papers/TreeGeodesicss/geodesics07.pdf. Last accessed July 31, 2018

work page 2007

[35] [35]

Conﬁdence sets for phylogenetic trees

Amy Willis. Conﬁdence sets for phylogenetic trees. Journal of the American Statistical Asso- ciation, pages 1–10, 2018

work page 2018

[36] [36]

Uncertainty in phylogenetic tree estimates

Amy Willis and Rayna Bell. Uncertainty in phylogenetic tree estimates. Journal of Computa- tional and Graphical Statistics , 27(3):542–552, 2018. 28

work page 2018