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Expected and minimal values of a universal tree balance index

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Pith's one-line read This paper establishes the tree balance index J1 as a universal measure by proving its leafy-tree identity, bounding its expected values, and characterizing its minimizers.

desk verdict Genuinely useful expected-value and extremal results for J1, but Proposition 3(ii)'s uniform-model bound of 0.057 is not established by the supplied proof, which only reaches 0.061. read the letter →

arxiv 2507.08615 v1 pith:27AIJR2X submitted 2025-07-11 q-bio.QM

Veselin Manojlović , Armaan Ahmed , Yannick Viossat , Robert Noble This is my paper · ORCID
classification q-bio.QM MSC 05C0505C9092D15
keywords treebalanceindexShannonentropySackinYulemodeluniformbroomweight-balancedHuffmancoding
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

J1 is a tree balance index defined as a weighted mean of Shannon-entropy node scores, applicable to any rooted tree with arbitrary outdegrees and node sizes. The paper argues that J1 is a genuinely universal index: it generalizes the ratio condition behind weight-balanced binary search trees, and the Huffman tree maximizes J1 among binary leafy trees. It proves that the natural approximation E(J1) ≈ n log2 n / E(IS) is accurate, with a Jensen gap below 0.008 for Yule-model trees (tending to 0) and below about 0.057 for uniform-model trees. The paper also analyzes the least balanced leafy trees: caterpillar trees minimize J1 only up to n = 4, and it gives asymptotic descriptions of the least balanced broom trees, conjecturing that brooms are the global minimizers.

What carries the argument

The central object is the leafy tree identity J1(T) = n log_m n / IS(T) (Proposition 1), which equates J1 with the harmonic-mean-normalized Sackin index for full m-ary leafy trees with unit leaves. This identity reduces expected-value questions to properties of the reciprocal of the Sackin index IS, letting the paper apply sharpened Jensen inequalities and known variance formulas, and it reduces extremal questions to minimizing IS. The second machinery is the broom tree family, with the explicit formula J1^B(n,k,p) = 2[kp + (kp + n − k) log2(kp + n − k) − kp log2(kp)] / ((2kp + n − k)(n − k + 1)), whose analysis yields the asymptotic minimizers and the phase transitions in the relative leaf size p.

What would settle it

Enumerate all leafy trees with, say, 13, 14, and 15 leaves and no outdegree-one nodes, compute J1 exactly for each, and compare the minimum against the best broom-tree value from the explicit formula; a single non-broom tree with a smaller J1 value would disprove Conjecture 5.

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Extended reading notes

Core claim

The central claim is that the leafy tree identity J1(T) = n log_m n / IS(T), valid for full m-ary leafy trees with equal unit leaves, turns J1 into a normalized reciprocal Sackin index. This identity explains the unification of biology and computer science: the weight-balanced tree's average entropy coincides with J1 in the binary uniform-leaf case, and Huffman coding, which minimizes weighted path length, maximizes J1. From this identity the paper proves tight uniform bounds on the Jensen gap between the true expected J1 and the arithmetic-mean approximation under the Yule and uniform models, and it establishes the asymptotic behavior of the minimizers. For leafy broom trees with equal leaves, the least balanced broom has most of its leaves in the head, with 1 − r* ∼ √2/√(log2 n); if the head leaves are smaller than the handle leaves (p ≤ 1/2), the caterpillar is asymptotically the minimizer; if p > 1/2, then 1 − r* ∼ p√2/√((2p − 1) log2 n). These broom-tree results are conditional on the unproven Conjecture 5 that brooms are the global minimizers, which the paper verifies exhaustively only for n ≤ 12.

Load-bearing premise

The paper's global-minimum conclusions rely on Conjecture 5, that a broom tree always minimizes J1 among leafy trees with equal leaves and no outdegree-one nodes; this is verified only up to 12 leaves, and until proved the detailed broom-tree asymptotics concern the broom family, not the global minimum.

Editorial extensions

If this is right

  • For Yule-model trees, E(J1) ≈ n log2 n / E(IS) has error below 0.008 for all n and the error tends to 0; asymptotically J1 → 1/(2 ln 2) ≈ 0.72 in probability.
  • For uniform-model trees, the same approximation has error below 4/(3πe²) ≈ 0.057 for all n, and numerical evidence suggests the true gap never exceeds 0.02.
  • J1 maximization on binary leafy trees is exactly Huffman coding, so entropy-optimal codes coincide with maximally balanced trees in this setting.
  • Among leafy bifurcating trees with equal leaves the caterpillar minimizes J1, but when outdegrees greater than 1 are allowed the caterpillar is optimal only for n ≤ 4; for equal leaves the least balanced broom tree has most leaves in the head, with 1 − r* ∼ √2/√(log2 n).
  • If head leaves are sufficiently smaller than handle leaves (p ≤ 1/2), the caterpillar is asymptotically the least balanced broom tree; if p > 1/2, the least balanced broom tree's head grows large, with 1 − r* ∼ p√2/√((2p − 1) log2 n).

Reading between the lines

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

  • If Conjecture 5 is proved, the broom-tree formulas would give the exact global minimum of J1 among leafy trees with equal leaves and no outdegree-one nodes, providing a rigorous null baseline for the maximally imbalanced trees in biology.
  • The equivalence with Huffman coding suggests that J1 can be read as a measure of how far a tree's leaf-weight distribution is from an entropy-optimal code, potentially connecting tree balance to coding-theoretic quantities such as code-tree cost.
  • The phase transition at p = 1/2 implies that fine-grained node-size information can qualitatively change the extremal tree topology, so empirical applications with uncertain leaf sizes should treat minimizer shape as sensitive to those estimates.
  • The authors' unproven assumption that E[X_n^{-1}] → E[A^{-1}] for the uniform model could be tested directly by simulation; if it holds, the uniform-model asymptotic E[J1] ∼ E[A^{-1}] n^{-1/2} log2 n decays polynomially rather than logarithmically.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 4 minor

Summary. The paper studies the universal tree balance index J1 for rooted trees with arbitrary node sizes. Its main contributions are: (1) establishing conceptual links between J1 and classical computer science notions such as weight-balanced trees and Huffman coding; (2) proving bounds on the Jensen gap J(n) between E[J1] and n log2 n / E[IS] under the Yule and uniform models; and (3) analyzing minimal values of J1 for leafy broom trees with equally sized leaves and with a two-scale leaf-size relaxation. The paper also formulates Conjecture 5, that the J1-minimizing leafy tree with no outdegree-1 nodes is a broom tree, and derives asymptotic results for minimizers within the broom family.

Significance. If the results were fully established, the paper would strengthen the case for J1 as a practically useful cross-disciplinary tree balance index: it would provide rigorous null-model reference values, connect J1 to known computer-science concepts, and characterize extremal tree shapes. The manuscript has genuine strengths: Proposition 3(i) gives explicit finite-n bounds for the Yule model, the exact rational tables (Tables A1 and A2) are valuable, the connection to Wong and Nievergelt's average entropy is interesting, and the broom-tree analysis is substantive. However, two load-bearing technical gaps, detailed below, prevent the paper from delivering on its central quantitative claims.

major comments (2)
  1. [Appendix A.1, Proposition 3(ii)] The proof does not establish the stated uniform-model bound. Equations (A6) through (A8) and the limits EU(IS) ~ sqrt(pi) n^(3/2) and VU(IS) ~ (10/3 - pi) n^3 imply that the upper bound on the Jensen gap tends to (10/3 - pi)/pi = 10/(3pi) - 1 ≈ 0.061, which is strictly larger than the claimed constant 4/(3 pi e^2) ≈ 0.057. Thus the argument as written supports only a weaker asymptotic bound, and Proposition 3(ii) as stated is unproven. The exact values in Table A2 are finite and do not cover all n, so they cannot close the gap.
  2. [Section 2.6, Proposition 4] The proof compares only the caterpillar tree (k = 2) with the broom tree having k = 3, using Eq. (12). It does not consider trees with maximal outdegree greater than 2, such as the star tree, even though such trees are included in the proposition's domain ('no nodes of outdegree 1'). Therefore the proof does not establish that the caterpillar minimizes J1 among all such trees for n ≤ 4; and for n > 4 it only shows that the caterpillar is not minimal within the broom subfamily, not that it fails to minimize J1 among all eligible trees. The global statement of Proposition 4 requires a broader argument.
minor comments (4)
  1. [Appendix A.1, Eq. (A3)] In the displayed upper bound, the factor n log2 n appears in both numerator and denominator; after cancellation the bound is simply VarY(IS)/mu_Y^2. Simplifying the expression would make the subsequent asymptotic comparison clearer.
  2. [Appendix A.2.2 and Figure 2] The uniform-model asymptotic approximation E[J1] ~ E[A^{-1}] n^{-1/2} log2 n is explicitly said to rest on a conjecture about the convergence of the inverse moment E[X_n^{-1}] to E[A^{-1}]. Please mark this conjecture in the main text near Figure 2, since the red curve in panel (D) is presented without this caveat.
  3. [General notation] The abstract uses J^1 while the body uses J1; please standardize the notation throughout.
  4. [Discussion] The sentence 'The errors in our approximations are small enough as to be negligible in many practical applications' should be qualified in light of Proposition 3(ii) not yet being proven; at present the uniform-model error bound is an open mathematical statement.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the expected and minimal J1 values are derived from the independently proved leafy tree identity and external theorems (Liao-Berg, Cardona et al., Rösler, Takács), with no fitted parameters renamed as predictions.

full rationale

The paper's central identity J1(T) = n log2 n / IS(T) is quoted from Lemant et al. (2022), where it is proved as a theorem; it is not defined in terms of the quantities being predicted. The null-model expectations are obtained by combining this identity with external formulas for the mean and variance of the Sackin index (Kirkpatrick and Slatkin; Cardona et al.) and with Liao and Berg's Jensen-gap inequality. No parameter is fitted to data and then called a prediction. The minimal-value results are derived from the closed-form expression for J1 on broom trees (Equation 12) and from analytical asymptotic arguments, not from fitting. The paper explicitly labels Conjecture 5 as unproven beyond n=12 and the uniform-model inverse-moment convergence as a conjecture on which an approximation is based; these are honest limitations rather than disguised inputs. The proof of Proposition 3(ii) appears numerically insufficient for the claimed constant, and Proposition 4's proof checks only broom trees rather than all leafy trees, but these are correctness or completeness gaps, not circularity. Self-citations are present, but the load-bearing cited result is a parameter-free theorem proved in the cited work, and the other cited results are external. There is no step where a claimed derivation is equivalent to its own input by construction.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The central claims rest on standard probabilistic bounds (Liao-Berg), known exact formulas for Sackin expectation and variance under Yule and uniform models, and asymptotic distribution results (Roesler, Takacs, Flajolet-Louchard). These are external, published results; no parameters are fitted to data in this paper.

assumptions (6)
  • standard math Liao-Berg sharpened Jensen inequality provides two-sided bounds on E[f(X)]-f(E[X]) in terms of the variance, assuming f is twice differentiable and X is bounded.
    Used in Appendix A.1 to bound the Jensen gap for both null models.
  • standard math Exact formulas for the expectation and variance of the Sackin index under Yule and uniform models, from Kirkpatrick and Slatkin, Cardona et al., and Mir et al.
    Used in Appendix A.1, equations A3 to A8; these are external published results.
  • standard math Roesler's convergence of the normalized Quicksort comparison count, and its moments, to a limiting distribution Y, used to derive the asymptotic error rate of Taylor approximations under the Yule model.
    Invoked in Appendix A.2.1 for moment estimates; absolute convergence in Wasserstein distance is asserted.
  • standard math Takacs's Airy distribution moment results and Flajolet-Louchard's formula for E[A^-1], used for the uniform-model asymptotic approximation.
    Invoked in Appendix A.2.2; the convergence of the inverse moment E[X_n^-1] is conjectured, not proven.
  • domain assumption The Yule and uniform models are the appropriate null models for expected tree balance comparisons; the analysis is restricted to leafy trees with equally sized leaves and no nodes of outdegree 1.
    Motivates the restriction of the minimal-value analysis to this family; the biological relevance of these constraints is standard in the tree balance literature.
  • ad hoc to paper Conjecture 5: among leafy trees with equally sized leaves and no nodes of outdegree 1, the J1-minimizer is a broom tree.
    Unproven but exhaustively verified for n up to 12; Propositions 6 through 11 describe least balanced broom trees, which characterize the global minimum only if this conjecture holds.

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Pith. "Pith review of Expected and minimal values of a universal tree balance index." pith.science (2026). https://pith.science/paper/27AIJR2X

@misc{pith2026250708615,
  author       = {Pith},
  title        = {Pith review of: Expected and minimal values of a universal tree balance index},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/27AIJR2X}},
  note         = {Machine review of arXiv:2507.08615}
}
abstract

Although the analysis of rooted tree shape has wide-ranging applications, notions of tree balance have developed independently in different domains. In computer science, a balanced tree is one that enables efficient updating and retrieval of data, whereas in biology tree balance quantifies bias in evolutionary processes. The lack of a precise connection between these concepts has stymied the development of universal indices and general results. We recently introduced a new tree balance index, $J^1$, that, unlike prior indices popular among biologists, permits meaningful comparison of trees with arbitrary degree distributions and node sizes. Here we explain how our new index generalizes a concept that underlies the definition of the weight-balanced tree, an important type of self-balancing binary search tree. Our index thus unifies the tree balance concepts of biology and computer science. We provide new analytical results to support applications of this universal index. First, we quantify the accuracy of approximations to the expected values of $J^1$ under two important null models: the Yule process and the uniform model. Second, we investigate minimal values of our index. These results help establish $J^1$ as a universal, cross-disciplinary index of tree balance that generalizes and supersedes prior approaches.

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Works this paper leans on

45 extracted references · 30 canonical work pages

  1. [1]

    : Stochastic models and descriptive statistics for phylogenetic trees, from Yule to today

    barticle Aldous , D.J. : Stochastic models and descriptive statistics for phylogenetic trees, from Yule to today . Statistical Science 16 ( 1 ), 23 -- 34 ( 2001 ) 10.1214/ss/998929474 barticle

    arXiv 2001

  2. [2]

    , Sack , J.-R

    barticle Atkinson , M.D. , Sack , J.-R. : Generating binary trees at random . Information Processing Letters 41 ( 1 ), 21 -- 23 ( 1992 ) 10.1016/0020-0190(92)90075-7 barticle

    work page doi:10.1016/0020-0190(92)90075-7 1992

  3. [3]

    , Fran c ois , O

    barticle Blum , M.G.B. , Fran c ois , O. : On statistical tests of phylogenetic tree imbalance: T he S ackin and other indices revisited . Mathematical Biosciences 195 ( 2 ), 141 -- 153 ( 2005 ) 10.1016/j.mbs.2005.03.003 barticle

    doi:10.1016/j.mbs.2005.03.003 2005

  4. [4]

    , François , O

    barticle Blum , M.G.B. , François , O. , Janson , S. : The mean, variance and limiting distribution of two statistics sensitive to phylogenetic tree balance . The Annals of Applied Probability 16 ( 4 ), 2195 -- 2214 ( 2006 ) 10.1214/105051606000000547 barticle

    doi:10.1214/105051606000000547 2006

  5. [5]

    , Schrago , C.G

    barticle Barzilai , L.P. , Schrago , C.G. : Signatures of natural selection in tree topology shape of serially sampled viral phylogenies . Molecular Phylogenetics and Evolution 183 , 107776 ( 2023 ) 10.1016/j.ympev.2023.107776 barticle

    work page arXiv 2023

  6. [6]

    , Sholokova , A

    barticle Bozic , I. , Sholokova , A. , Kaveh , K. : Neoantigen evolution and response to checkpoint inhibitor immunotherapy in colorectal cancer . Research Square ( 2024 ) 10.21203/rs.3.rs-4922340/v1 barticle

    work page doi:10.21203/rs.3.rs-4922340/v1 2024

  7. [7]

    , Fischer , M

    barticle Coronado , T.M. , Fischer , M. , Herbst , L. , Rossell \'o , F. , Wicke , K. : On the minimum value of the C olless index and the bifurcating trees that achieve it . Journal of Mathematical Biology 80 , 1993 -- 2054 ( 2020 ) 10.1007/s00285-020-01488-9 barticle

    doi:10.1007/s00285-020-01488-9 1993

  8. [8]

    , Gardy , J

    barticle Colijn , C. , Gardy , J. : Phylogenetic tree shapes resolve disease transmission patterns . Evolution, Medicine, and Public Health 2014 ( 1 ), 96 -- 108 ( 2014 ) 10.1093/emph/eou018 barticle

    work page doi:10.1093/emph/eou018 2014

Show all 45 references
  1. [9]

    , Hayati , M

    barticle Chindelevitch , L. , Hayati , M. , Poon , A.F. , Colijn , C. : Network science inspires novel tree shape statistics . PLOS One 16 ( 12 ), 0259877 ( 2021 ) 10.1371/journal.pone.0259877 barticle

    2021 doi

  2. [10]

    , Mir , A

    barticle Cardona , G. , Mir , A. , Rossell \'o , F. : Exact formulas for the variance of several balance indices under the Yule model . Journal of Mathematical Biology 67 ( 6 ), 1833 -- 1846 ( 2013 ) 10.1007/s00285-012-0615-9 barticle

    2013 doi

  3. [11]

    : Review of Phylogenetics , The Theory and Practice of Phylogenetic Systematics

    barticle Colless , D.H. : Review of Phylogenetics , The Theory and Practice of Phylogenetic Systematics . Systematic Zoology 31 ( 1 ), 100 -- 104 ( 1982 ) 10.2307/2413420 barticle

    1982 doi

  4. [12]

    , Campos , P.R

    botherref Freitas , O. , Campos , P.R. , Araujo , S.B. : Patch biogeography under intermittent barriers: macroevolutionary consequences of microevolutionary processes. Journal of Evolutionary Biology, 035 (2024) 10.1093/jeb/voae035 botherref

    2024 doi

  5. [13]

    , Gao , Y

    botherref Feder , A.F. , Gao , Y. : Detecting branching rate heterogeneity in multifurcating trees with applications in lineage tracing data. bioRxiv, 2024--06 (2024) 10.1101/2024.06.27.601073 botherref

    2024 doi

  6. [14]

    , Gittenberger , B

    barticle Fuchs , M. , Gittenberger , B. : Sackin indices for labeled and unlabeled classes of galled trees . Journal of Mathematical Biology 90 ( 4 ), 1 -- 34 ( 2025 ) 10.1007/s00285-025-02205-0 barticle

    2025 doi

  7. [15]

    , Herbst , L

    bbook Fischer , M. , Herbst , L. , Kersting , S. , K \"u hn , A.L. , Wicke , K. : Tree Balance Indices: A Comprehensive Survey . Springer , Cham, Switzerland ( 2023 ). 10.1007/978-3-031-39800-1 bbook

    2023 doi

  8. [16]

    : Extremal values of the Sackin tree balance index

    barticle Fischer , M. : Extremal values of the Sackin tree balance index . Annals of Combinatorics 25 ( 2 ), 515 -- 541 ( 2021 ) 10.1007/s00026-021-00539-2 barticle

    2021 doi

  9. [17]

    , Louchard , G

    barticle Flajolet , P. , Louchard , G. : Analytic variations on the A iry distribution . Algorithmica 31 ( 3 ), 361 -- 377 ( 2001 ) 10.1007/s00453-001-0056-0 barticle

    2001 doi

  10. [18]

    : The properties of random trees

    barticle Halton , J.H. : The properties of random trees . Information Sciences 47 ( 2 ), 95 -- 133 ( 1989 ) 10.1016/0020-0255(89)90009-1 barticle

    1989 doi

  11. [19]

    : Patterns in tree balance among cladistic, phenetic, and randomly generated phylogenetic trees

    barticle Heard , S.B. : Patterns in tree balance among cladistic, phenetic, and randomly generated phylogenetic trees . Evolution 46 ( 6 ), 1818 -- 1826 ( 1992 ) 10.1111/j.1558-5646.1992.tb01171.x barticle

    1992

  12. [20]

    : A method for the construction of minimum-redundancy codes

    barticle Huffman , D.A. : A method for the construction of minimum-redundancy codes . Proceedings of the IRE 40 ( 9 ), 1098 -- 1101 ( 1952 ) 10.1109/JRPROC.1952.273898 barticle

    1952

  13. [21]

    : The Q uicksort algorithm and related topics

    barticle Iliopoulos , V. : The Q uicksort algorithm and related topics . arXiv ( 2015 ) 10.48550/arXiv.1503.02504 https://arxiv.org/abs/1503.02504 1503.02504 barticle

    2015 doi arXiv

  14. [22]

    : The Art of Computer Programming, Volume 1: Fundamental Algorithms , 3 rd edn

    bbook Knuth , D.E. : The Art of Computer Programming, Volume 1: Fundamental Algorithms , 3 rd edn. Addison-Wesley Professional , Boston ( 1997 ) bbook

    1997

  15. [23]

    : The Art of Computer Programming, Volume 3: Sorting and Searching , 2 nd edn

    bbook Knuth , D.E. : The Art of Computer Programming, Volume 3: Sorting and Searching , 2 nd edn. Addison-Wesley Professional , Boston ( 1998 ) bbook

    1998

  16. [24]

    , Slatkin , M

    barticle Kirkpatrick , M. , Slatkin , M. : Searching for evolutionary patterns in the shape of a phylogenetic tree . Evolution 47 ( 4 ), 1171 -- 1181 ( 1993 ) 10.1111/j.1558-5646.1993.tb02144.x barticle

    1993

  17. [25]

    , Berg , A

    barticle Liao , J.G. , Berg , A. : Sharpening J ensen's inequality . The American Statistician 73 ( 3 ), 278 -- 281 ( 2019 ) 10.1080/00031305.2017.1419145 barticle

    2019 arXiv

  18. [26]

    o ni , J. , Cellerai , C. , Klimkait , T. , G \

    barticle Leventhal , G.E. , Kouyos , R. , Stadler , T. , Von Wyl , V. , Yerly , S. , B \"o ni , J. , Cellerai , C. , Klimkait , T. , G \"u nthard , H.F. , Bonhoeffer , S. : Inferring epidemic contact structure from phylogenetic trees . PLoS Computational Biology 8 ( 3 ), 10024...

    2012 doi

  19. [27]

    , Le Sueur , C

    barticle Lemant , J. , Le Sueur , C. , Manojlović , V. , Noble , R. : Robust, universal tree balance indices . Systematic Biology 71 ( 5 ), 1210 -- 1224 ( 2022 ) 10.1093/sysbio/syac027 barticle

    2022 doi

  20. [28]

    , Heard , S.B

    barticle Mooers , A.O. , Heard , S.B. : Inferring evolutionary process from phylogenetic tree shape . The Quarterly Review of Biology 72 ( 1 ), 31 -- 54 ( 1997 ) 10.1086/419657 barticle

    1997 doi

  21. [29]

    , Rosselló , F

    barticle Mir , A. , Rosselló , F. , Rotger , L. : A new balance index for phylogenetic trees . Mathematical Biosciences 241 ( 1 ), 125 -- 136 ( 2013 ) 10.1016/j.mbs.2012.10.005 barticle

    2013 doi

  22. [30]

    , Steel , M

    barticle McKenzie , A. , Steel , M. : Distributions of cherries for two models of trees . Mathematical Biosciences 164 ( 1 ), 81 -- 92 ( 2000 ) 10.1016/s0025-5564(99)00060-7 barticle

    doi

  23. [31]

    , Burri , D

    barticle Noble , R. , Burri , D. , Le Sueur , C. , Lemant , J. , Viossat , Y. , Kather , J.N. , Beerenwinkel , N. : Spatial structure governs the mode of tumour evolution . Nature Ecology & Evolution 6 ( 2 ), 207 -- 217 ( 2022 ) 10.1038/s41559-021-01615-9 barticle

    2022 doi

  24. [32]

    , Pradels , J

    barticle Nievergelt , J. , Pradels , J. , Wong , C.K. , Yue , P.C. : Bounds on the weighted path length of binary trees . Information Processing Letters 1 ( 6 ), 220 -- 225 ( 1972 ) 10.1016/0020-0190(72)90015-4 barticle

    1972 doi

  25. [33]

    , Reingold , E.M

    bchapter Nievergelt , J. , Reingold , E.M. : Binary search trees of bounded balance . In: Proceedings of the Fourth Annual ACM Symposium on Theory of Computing - STOC '72 , pp. 137 -- 142 . ACM Press , Denver, Colorado ( 1972 ). 10.1145/800152.804906 bchapter

    1972

  26. [34]

    , Verity , K

    barticle Noble , R. , Verity , K. : A new universal system of tree shape indices . bioRxiv ( 2023 ) 10.1101/2023.07.17.549219 barticle

    2023 doi

  27. [35]

    , Agapow , P.-M

    barticle Purvis , A. , Agapow , P.-M. : Phylogeny imbalance: taxonomic level matters . Systematic Biology 51 ( 6 ), 844 -- 854 ( 2002 ) 10.1080/10635150290102546 barticle

    2002 doi

  28. [36]

    : Tree thinking, time and topology: comments on the interpretation of tree diagrams in evolutionary/phylogenetic systematics

    barticle Podani , J. : Tree thinking, time and topology: comments on the interpretation of tree diagrams in evolutionary/phylogenetic systematics . Cladistics 29 ( 3 ), 315 -- 327 ( 2013 ) 10.1111/j.1096-0031.2012.00423.x barticle

    2013

  29. [37]

    : Vicariant patterns and historical explanation in biogeography

    barticle Rosen , D.E. : Vicariant patterns and historical explanation in biogeography . Systematic Zoology 27 ( 2 ), 159 -- 188 ( 1978 ) 10.2307/2412970 barticle

    1978 doi

  30. [38]

    : A limit theorem for ``quicksort"

    barticle Rösler , U. : A limit theorem for ``quicksort" . RAIRO-Theor. Inf. Appl. 25 ( 1 ), 85 -- 100 ( 1991 ) 10.1051/ita/1991250100851 barticle

    1991

  31. [39]

    Good ” and “bad

    barticle Sackin , M.J. : “ Good ” and “bad” phenograms . Systematic Biology 21 ( 2 ), 225 -- 226 ( 1972 ) 10.1093/sysbio/21.2.225 barticle

    1972 doi

  32. [40]

    , McKenzie , A

    botherref Steel , M. , McKenzie , A. : Properties of phylogenetic trees generated by Yule -type speciation models q. Mathematical Biosciences, 22 (2001) 10.1016/s0025-5564(00)00061-4 botherref

    2001 doi

  33. [41]

    , Maini , P.K

    barticle Scott , J.G. , Maini , P.K. , Anderson , A.R.A. , Fletcher , A.G. : Inferring Tumor Proliferative Organization from Phylogenetic Tree Measures in a Computational Model . Systematic Biology 69 ( 4 ), 623 -- 637 ( 2020 ) 10.1093/sysbio/syz070 barticle

    2020 doi

  34. [42]

    , Sokal , R.R

    barticle Shao , K.-T. , Sokal , R.R. : Tree Balance . Systematic Zoology 39 ( 3 ), 266 ( 1990 ) 10.2307/2992186 barticle

    1990 doi

  35. [43]

    : A B ernoulli excursion and its various applications

    barticle Takács , L. : A B ernoulli excursion and its various applications . Advances in Applied Probability 23 ( 3 ), 557 -- 585 ( 1991 ) 10.2307/1427622 barticle

    1991 doi

  36. [44]

    , Nievergelt , J

    barticle Wong , C.K. , Nievergelt , J. : Upper bounds for the total path length of binary trees . Journal of the ACM 20 ( 1 ), 1 -- 6 ( 1973 ) 10.1145/321738.321739 barticle

    1973

  37. [45]

    barticle Yule , G.U. : II . --- A mathematical theory of evolution, based on the conclusions of Dr . J . C . Willis, F. R. S . Philosophical Transactions of the Royal Society B 213 ( 402 ), 21 -- 87 ( 1925 ) 10.1098/rstb.1925.0002 barticle

    1925

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