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%I #12 Jul 29 2019 12:24:26
%S 4,34,368,4685,66416,1013268,16279788,271594611,4660794200,
%T 81747301898,1458812278424,26400987754054,483374731032868,
%U 8936983620559660,166617056922535080,3128790129161470470,59124052722375912960,1123458655726125274620,21452847767668402271220
%N Number of evolutionary duplication-loss-histories of the complete binary species tree with 4 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 complete binary species tree S of size k is a complete binary tree with k leaves. 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.
%F G.f.: 1/2-(1/2)*sqrt(1+6*sqrt(-5+6*sqrt(1-4*z)+4*z)-6*sqrt(1-4*z)-4*z).
%e The complete binary species tree with 4 leaves is equal to
%e a
%e / \
%e b c
%e / \ / \
%e 1 2 3 4
%e For convenience the internal nodes are labeled by a,b,c and the leaves by 1,2,3,4. The associated nodes in the histories will be denoted by the same labels.
%e The a(1)=4 histories with n=1 leaf are created by the following growth process:
%e a a a a
%e / / \ \
%e b b c c
%e / \ / \
%e 1 2 3 4
%e after two loss events each.
%o (PARI) z='z+O('z^20); Vec(1/2-(1/2)*sqrt(1+6*sqrt(-5+6*sqrt(1-4*z)+4*z)-6*sqrt(1-4*z)-4*z)) \\ _Jianing Song_, Jul 29 2019
%Y Cf. A000108 (caterpillar/complete binary species tree with 1 leaf, ordinary binary trees).
%Y Cf. A307696, A307697, A307698, A307700 (caterpillar species tree with 2, 3, 4, 5 leaves).
%K nonn
%O 1,1
%A _Cedric Chauve_, May 07 2019
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