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%I #32 Jul 31 2019 11:12:49
%S 2,7,34,200,1318,9354,69864,541323,4310950,35066384,290081932,
%T 2432766082,20635672664,176727482860,1526000459400,13270616752680,
%U 116124930068670,1021736927603190,9033726534916920,80220639767921370,715166816624282820,6398357633173869600
%N Number of evolutionary duplication-loss-histories with n leaves of the caterpillar species tree with 2 leaves.
%C An evolutionary history of size n is an ordered rooted (incomplete) binary tree with n leaves describing the evolution of a gene family of a species in phylogenomics. The caterpillar species tree S of size k is a binary tree with k leaves, where any left child is a leaf. Any node of the history is associated to a unique node of S, where specifically every leaf is associated to a leaf of S. A history is created by the following process (note that intermediate trees in this process may not be valid histories): Start with a root node associated to the root of S. For a given tree in the growth process, choose a leaf and perform a duplication, speciation, or (speciation-)loss event. A duplication event creates two children both associated to the same node as its parent. A speciation or (speciation-)loss event can only occur if the node is associated to an internal node in S. In that case, a speciation event creates two children associated to the children of the node in S. A (speciation-)loss event creates only a left or right child, associated to the left or right child in S, respectively.
%H C. Chauve, Y. Ponty, M. Wallner, <a href="https://arxiv.org/abs/1905.04971">Counting and sampling gene family evolutionary histories in the duplication-loss and duplication-loss-transfer models</a>, arXiv preprint arXiv:1905.04971 [math-CO], 2019.
%F G.f.: 1/2 - (1/2)*sqrt(-5 + 6*sqrt(1-4*z) + 4*z).
%e The caterpillar species tree with 2 leaves is equal to
%e a
%e / \
%e 1 2
%e For convenience the internal node is labeled by a, and the leaves by 1,2. The associated nodes in the histories will be denoted by the same labels.
%e The a(1)=2 histories with n=1 leaf are created by the following growth process:
%e a a
%e / \
%e 1 2
%e after one loss event each.
%e The a(2)=7 histories with n=2 leaves are created by the following growth process:
%e a a a a a a a
%e / \ / \ / \ / \ / \ / \
%e 1 2 1 2 a a a a a a a a
%e / \ / \ / / / \ \ \ \ /
%e 1 1 2 2 1 1 1 2 2 2 2 1
%o (PARI) my(z='z+O('z^30)); Vec(1/2-(1/2)*sqrt(-5+6*sqrt(1-4*z)+4*z)) \\ _Michel Marcus_, Apr 22 2019
%Y Caterpillar species tree sequences: A000108 (1 leaf), A307697 (3 leaves), A307698 (4 leaves), A307700 (5 leaves).
%K nonn
%O 1,1
%A _Michael Wallner_, Apr 22 2019
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