Generating normal networks via leaf insertion and nearest neighbor interchange
Abstract Background Galled trees are studied as a recombination model in theoretical population genetics. This class of phylogenetic networks has been generalized to tree-child networks and other network classes by relaxing a structural condition imposed on galled trees. Although these networks are...
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doaj-c26161990c00475a83894fe7adccbe1c2020-12-20T12:42:12ZengBMCBMC Bioinformatics1471-21052019-12-0120S201910.1186/s12859-019-3209-3Generating normal networks via leaf insertion and nearest neighbor interchangeLouxin Zhang0Department of Mathematics, National University of SingaporeAbstract Background Galled trees are studied as a recombination model in theoretical population genetics. This class of phylogenetic networks has been generalized to tree-child networks and other network classes by relaxing a structural condition imposed on galled trees. Although these networks are simple, their topological structures have yet to be fully understood. Results It is well-known that all phylogenetic trees on n taxa can be generated by the insertion of the n-th taxa to each edge of all the phylogenetic trees on n−1 taxa. We prove that all tree-child (resp. normal) networks with k reticulate nodes on n taxa can be uniquely generated via three operations from all the tree-child (resp. normal) networks with k−1 or k reticulate nodes on n−1 taxa. Applying this result to counting rooted phylogenetic networks, we show that there are exactly (2n)!2n(n−1)!−2n−1n! $\frac {(2n)!}{2^{n} (n-1)!}-2^{n-1} n!$ binary phylogenetic networks with one reticulate node on n taxa. Conclusions The work makes two contributions to understand normal networks. One is a generalization of an enumeration procedure for phylogenetic trees into one for normal networks. Another is simple formulas for counting normal networks and phylogenetic networks that have only one reticulate node.https://doi.org/10.1186/s12859-019-3209-3Tree-child networksnormal networks |
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DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Louxin Zhang |
spellingShingle |
Louxin Zhang Generating normal networks via leaf insertion and nearest neighbor interchange BMC Bioinformatics Tree-child networks normal networks |
author_facet |
Louxin Zhang |
author_sort |
Louxin Zhang |
title |
Generating normal networks via leaf insertion and nearest neighbor interchange |
title_short |
Generating normal networks via leaf insertion and nearest neighbor interchange |
title_full |
Generating normal networks via leaf insertion and nearest neighbor interchange |
title_fullStr |
Generating normal networks via leaf insertion and nearest neighbor interchange |
title_full_unstemmed |
Generating normal networks via leaf insertion and nearest neighbor interchange |
title_sort |
generating normal networks via leaf insertion and nearest neighbor interchange |
publisher |
BMC |
series |
BMC Bioinformatics |
issn |
1471-2105 |
publishDate |
2019-12-01 |
description |
Abstract Background Galled trees are studied as a recombination model in theoretical population genetics. This class of phylogenetic networks has been generalized to tree-child networks and other network classes by relaxing a structural condition imposed on galled trees. Although these networks are simple, their topological structures have yet to be fully understood. Results It is well-known that all phylogenetic trees on n taxa can be generated by the insertion of the n-th taxa to each edge of all the phylogenetic trees on n−1 taxa. We prove that all tree-child (resp. normal) networks with k reticulate nodes on n taxa can be uniquely generated via three operations from all the tree-child (resp. normal) networks with k−1 or k reticulate nodes on n−1 taxa. Applying this result to counting rooted phylogenetic networks, we show that there are exactly (2n)!2n(n−1)!−2n−1n! $\frac {(2n)!}{2^{n} (n-1)!}-2^{n-1} n!$ binary phylogenetic networks with one reticulate node on n taxa. Conclusions The work makes two contributions to understand normal networks. One is a generalization of an enumeration procedure for phylogenetic trees into one for normal networks. Another is simple formulas for counting normal networks and phylogenetic networks that have only one reticulate node. |
topic |
Tree-child networks normal networks |
url |
https://doi.org/10.1186/s12859-019-3209-3 |
work_keys_str_mv |
AT louxinzhang generatingnormalnetworksvialeafinsertionandnearestneighborinterchange |
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