A307941 - OEIS

%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