, If we assume an augmentation of type T , it is not difficult to see that an optimal choice for ? 1 and its corresponding sequence ? T are respectively ? and AAAC

, If we explore the different possibilities for an augmentation of type H, we find that the best choice is {? 1 , ? 2 } = {?, ?} with sequences AAAB and BBCC and with value ?

, Since µ < 1/3, we get that ?(?) > ?({?, ?}) and we opt for the augmentation N 0 = T ?1 (N, ?; {?}, ?) shown in Fig, p.17

, Taking any node y ? C different from ? we get that ?(y) < 0 and hence they are not good candidates. If we take y = ?, taking into account that µ < 1/3, we find an optimal sequence ?(?) = AACC. The value of ?(?) corresponds to three different evolution processes: two mutations, from AAAC to AACC and from AACC to AACC, and a hybridization of the sequences AACC and BBBB to BBCC, and hence we get ?(?) = (1 ? 2µ) 7 µ/2. Now, this value must be compared with the probability of evolution without this hybridization, i.e. the probability of speciations from BBBB to BBCC and from AAAC to AACC which is µ 3 (1 ? 2µ) 5 . Since (1 ? 2µ) 7 µ/2 > µ 3 (1 ? 2µ) 5 (assuming that µ < 0.2928, N 0 ) of ? and ? are in C

P. Gambette, L. Van-iersel, M. Jones, M. Lafond, F. Pardi et al., Celine Rearrangement Moves on Rooted Phylogenetic Networks, PLoS Computational Biology, vol.8, issue.13, p.1005611, 2017.

W. P. Maddison, Gene Trees in Species Trees. Systematic Biology, vol.46, issue.3, pp.523-536, 1997.

C. Scornavacca and D. H. Huson, A Survey of Combinatorial Methods for Phylogenetic Networks, Genome Biology and Evolution, vol.3, pp.23-35, 2010.
URL : https://hal.archives-ouvertes.fr/hal-02155011

C. Semple, M. Baroni, and M. Steel, Hybrids in Real Time, Systematic Biology, vol.55, issue.1, pp.46-56, 2006.

M. Baroni, C. Semple, and M. Steel, A Framework for Representing Reticulate Evolution, Annals of Combinatorics, vol.8, issue.4, pp.391-408, 2005.

P. L. Erdos, C. Semple, and M. Steel, A class of phylogenetic networks reconstructable from ancestral profiles, 2019.

D. Gusfield, S. Eddhu, and C. Langley, Efficient Reconstruction of Phylogenetic Networks with Constrained Recombination, Proceedings of the IEEE Computer Society Conference on Bioinformatics. CSB '03, p.363, 2003.

D. H. Huson and T. H. Klöpper, Beyond Galled Trees -Decomposition and Computation of Galled Networks, Research in Computational Molecular Biology, pp.211-225, 2007.

L. Van-iersel and S. Kelk, Constructing the Simplest Possible Phylogenetic Network from Triplets, Algorithmica, vol.60, issue.2, pp.207-235, 2011.

G. Cardona, F. Rosselló, and G. Valiente, Comparison of Tree-Child Phylogenetic Networks, IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol.6, issue.4, pp.552-569, 2009.

A. R. Francis and M. Steel, Which phylogenetic networks are merely trees with additional arcs? Systematic biology, vol.64, pp.768-777, 2015.

G. Cardona, J. C. Pons, and F. Rosselló, A reconstruction problem for a class of phylogenetic networks with lateral gene transfers, Algorithms for Molecular Biology, vol.10, issue.1, p.28, 2015.

L. Van-iersel, C. Semple, and M. Steel, Locating a tree in a phylogenetic network, Information Processing Letters, vol.110, issue.23, pp.1037-1043, 2010.

L. Van-iersel and V. Moulton, Trinets encode tree-child and level-2 phylogenetic networks, Journal of Mathematical Biology, vol.68, issue.7, pp.1707-1729, 2014.

C. Semple, Phylogenetic Networks with Every Embedded Phylogenetic Tree a Base Tree, Bulletin of Mathematical Biology, vol.78, issue.1, pp.132-137, 2016.

M. Bordewich, C. Semple, and N. Tokac, Constructing Tree-Child Networks from Distance Matrices. Algorithmica, vol.80, pp.2240-2259, 2018.

F. Pardi and C. Scornavacca, Reconstructible phylogenetic networks: do not distinguish the indistinguishable, PLoS computational biology, vol.11, issue.4, p.1004135, 2015.
URL : https://hal.archives-ouvertes.fr/lirmm-01194638

A. D. Gunawan, J. Rathin, and L. Zhang, Counting and Enumerating Galled Networks, 2018.

C. Mcdiarmid, C. Semple, and D. Welsh, Counting Phylogenetic Networks, Annals of Combinatorics, vol.19, issue.1, pp.205-224, 2015.

M. Fuchs, B. Gittenberger, and M. Mansouri, Counting Phylogenetic Networks with Few Reticulation Vertices: Tree-Child and Normal Networks. arXiv e-prints, 2018.

G. Cardona, M. Llabrés, F. Rosselló, and G. Valiente, A distance metric for a class of tree-sibling phylogenetic networks, Bioinformatics, vol.24, issue.13, pp.1481-1488, 2008.

G. Cardona, M. Llabrés, F. Rosselló, and G. Valiente, Metrics for phylogenetic networks II: Nodal and triplets metrics, IEEE/ACM Transactions on Computational Biology and Bioinformatics, vol.6, issue.3, pp.454-469, 2009.

J. Oldman, T. Wu, L. Van-iersel, and V. Moulton, Trilonet: piecing together small networks to reconstruct reticulate evolutionary histories. Molecular biology and evolution, vol.33, pp.2151-2162, 2016.

K. T. Huber, L. Van-iersel, V. Moulton, C. Scornavacca, and T. Wu, Reconstructing phylogenetic level-1 networks from nondense binet and trinet sets, Algorithmica, vol.77, issue.1, pp.173-200, 2017.
URL : https://hal.archives-ouvertes.fr/hal-02154892

G. Jin, L. Nakhleh, S. Snir, and T. Tuller, Maximum likelihood of phylogenetic networks, Bioinformatics, vol.22, issue.21, pp.2604-2611, 2006.

C. Meng and L. S. Kubatko, Detecting hybrid speciation in the presence of incomplete lineage sorting using gene tree incongruence: a model. Theoretical population biology, vol.75, pp.35-45, 2009.

D. Gusfield, ReCombinatorics: the algorithmics of ancestral recombination graphs and explicit phylogenetic networks, 2014.

G. Cardona and D. Sánchez, PhyloNetworks: A Python library for phylogenetic networks, 2012.