OFFSET
1,1
COMMENTS
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.
LINKS
Sean A. Irvine, Java program (github)
FORMULA
G.f.: 1/2-(1/2)*sqrt(1-6*v+6*w+6*u-6*t-4*z) where t = sqrt(1-4*z), u = sqrt(-5+6*t+4*z), v = sqrt(1+6*u-6*t-4*z) and w = sqrt(-5+6*v-6*u+6*t+4*z)
EXAMPLE
See A307941 (complete binary species tree with 4 leaves).
CROSSREFS
Cf. A000108 (caterpillar/complete binary species tree with 1 leaf, ordinary binary trees).
KEYWORD
nonn
AUTHOR
Cedric Chauve, May 07 2019
STATUS
approved