Supplementary Material, corresponding to the manuscript Accumulated Coalescence Rank and Excess Gene count for Species Tree Inference

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1 Supplementary Material, corresponding to the manuscript Accumulated Coalescence Rank and Excess Gene count for Species Tree Inference Sourya Bhattacharyya and Jayanta Mukherjee Department of Computer Science and Engineering Indian Institute of Technology Kharagpur Kharagpur, West Bengal 7130, India January, Proof of lemmas Lemma 1 min x,y R G (x, y). Proof: From denition, min x,y R G (x, y) =, when x and y are siblings, and are placed at the lowest level of a caterpillar. Here, R G (x, y) is also. N Lemma For a perfect caterpillar G covering N taxa, max x,y R G (x, y) = i= i = N(N+1) - 1. Proof: As R G (x, y) N for any couplet (x, y), maximum R G (x, y) occurs when it includes all the rank values between to N. It corresponds to the case when x (or y) lies at the lowest level of the caterpillar, and LCA G xy is the root of the caterpillar. Example couplets (A,E) and (A,F) in the tree of Fig. exhibit maximum accumulated rank values. N 1 Lemma 3 For non-caterpillar trees covering N taxa, max x,y R G (x, y) = { N+1 lg N i} + N. Proof: Maximum R G (x, y) includes λ G(r) of the root r in G. Further, it should include the coalescence ranks of the internal nodes lying at the lowest levels of the either side of the root. For a height balanced complete binary gene tree, coalescence rank of each of the lowest level internal nodes is N - ( lg N - 1) = N lg N. So, maximum R G (x, y) will twice include the values from (N lg N ) to (N - N 1 1), and also include N, thus equaling { i} + N. N+1 lg N Lemma and lemma 3 show that maximum value of R is O (()N ) (given L(G) = N). Next, we nd the number of distinct R G (x, y) values in all couplets (x, y) of a gene tree G, to derive the exact correspondence between the number of couplets and R. Lemma 4 For a gene tree G with N taxa, maximum number of distinct R G (x, y) N 3N+4. Proof: For any three taxa x, y, z in G, R G (x, y) = R G (x, z), means that y and z are siblings. So, to achieve maximum number of distinct R G (x, y) values, we require to minimize the number of sibling taxa. Such minimization happens for a caterpillar, with just one pair of sibling taxa. Here, (N ) is the number of couplets having duplicate R entries, and the number of distinct R G (x, y) values is N(N 1) (N ) = N 3N+4. Lemma 5 For a binary gene tree G with N taxa, minimum number of distinct R G (x, y) values N +4N 3. 1

2 Proof: Lemma 4 shows that to achieve minimum number of distinct R G (x, y) values, we require maximizing the number of sibling taxa pairs. Given L(G) = N, maximum number of sibling couplets is (N/). For minimum distinct rank values, G needs to be symmetrical, indicating (N/4) couplets on either side of the root. Case 1: Couplets C1 and C are on the same side of the root. There will be N 4 distinct rank values. Case : Couplets C1 and C are on either side of the root. Individual such pair (C1, C) can be chosen in ( ) N 4 ways, each of which have a distinct rank value. Considering both cases, minimum number of distinct R G (x, y) values is ( N 4 ) + N 4 = N +4N 3.

3 Figure 1: Topology and bootstrap clade supports for the Angiosperm dataset, for NJst and STAR. Red branch corresponds to the topology from STAR. 100% bootstrap support values are indicated by `*' symbol. Topology of a group of taxa, depicting 100% bootstrap, are represented by the group name, shown in colored and underlined labels. 3

4 Figure : Topology and bootstrap clade supports for the Angiosperm dataset, for ASTRAL and mulrf. Red branch corresponds to the topology from mulrf. 100% bootstrap support values are indicated by `*' symbol. Topology of a group of taxa, depicting 100% bootstrap, are represented by the group name, shown in colored and underlined labels. 4

5 RAxML command used for Bootstrapping RAxML [6] was used to generate the bootstrap replicates for biological datasets. We followed [4] for such bootstrapping. command for bestml input tree: raxmlhpc-sse3 -m GTRGAMMA -s alignmentle -n bestml -N 0 -p pseed command for bootstrapping: raxmlhpc-sse3 -m GTRGAMMA -s alignmentle -n BSreplica -N 00 -p pseed -b bseed 5

6 (a) Species tree derived by Phylonet [7] (b) Species tree derived by igtp [1] Figure 3: Species trees obtained by Phylonet and igtp, for Amniota [3, ] dataset. 6

7 Figure 4: Detailed view of the topology and bootstrap clade supports corresponding to the Mammalian dataset, for AcRNJXL and the reference approaches. Species tree topologies and corresponding bootstrap support values (less than 100) are shown in the order of ASTRAL/mulRF/STAR/AcRNJXL. The method NJst produces identical performance with STAR. 7

8 Figure 5: Species trees obtained by Phylonet[7], for Mammalian [3, 5] dataset. 8

9 Figure 6: Species trees obtained by igtp [1], for Mammalian [3, 5] dataset. 9

10 Figure 7: Species trees obtained by igtp [1] for the Angiosperm dataset [8]. Bootstrap clade supports are also shown. 10

11 References [1] R. Chaudhary, M. S. Bansal, A. Wehe, D. Fernández-Baca, and O. Eulenstein. igtp: a software package for large-scale gene tree parsimony analysis. BMC Bioinformatics., 3(574):17, 010. [] Y. Chiari, V. Cahais, N. Galtier, and F. Delsuc. Phylogenomic analyses support the position of turtles as the sister group of birds and crocodiles (archosauria). BMC Biology, 10(65):114, 01. [3] S. Mirarab, R. Reaz, M. S. Bayzid, T. Zimmermann, M. S. Swenson, and T. Warnow. Astral: genomescale coalescent-based species tree estimation. Bioinformatics, 30(17):i541i548, 014. [4] S. Mirarab and T. Warnow. Astral-ii: coalescent-based species tree estimation with many hundreds of taxa and thousands of genes. Bioinformatics, 31(1):i44i5, 015. [5] S. Song, L. Liu, S. V. Edwards, and S. Wu. Resolving conict in eutherian mammal phylogeny using phylogenomics and the multispecies coalescent model. Proc Natl Acad Sci USA., 109(37): , 01. [6] A. Stamatakis. Raxml-vi-hpc: maximum likelihood-based phylogenetic analyses with thousands of taxa and mixed models. Bioinformatics, (1):688690, 006. [7] Cuong Than and Luay Nakhleh. Species tree inference by minimizing deep coalescences. PLOS Computational Biology, 5(9):11, 009. [8] Z. Xi, L. Liu, J. S. Rest, and C. C. Davis. Coalescent versus concatenation methods and the placement of amborella as sister to water lilies. Syst. Biol., 63(6):91993,