Skip to main content

Table 1 Notation used in the paper

From: A polynomial time algorithm for calculating the probability of a ranked gene tree given a species tree

Symbol

Meaning

T

Species tree with real-valued divergence times

G

Ranked gene tree (real-valued coalescence times not specified)

n

The number of leaves of T and G

s i

Speciation times, with s1 >> sn−1, let s0 =

τ i

Intervals between speciation times, τ i = [s i ,si−1)

i

The number of gene tree lineages at time s i

m i

The number of coalescence events in interval τ i

G i , i

The ranked gene tree observed from time 0 to time s i

g i

The minimum number of gene tree lineages at time s i

y i,z

Population z in interval τ i in beaded tree

u i

Internal node (coalescence) with rank i in the gene tree, u1 is most ancient, un−1 is the most recent

k i,j,z

The number of lineages available for coalescence in population yi,z just after the j th coalescence (considered forward in time) in interval τ i ; ki,0,z is the number of lineages “exiting” at time si−1

δ(y),δ(u)

The set of leaves descended from a node of the species tree or gene tree, respectively

lca(u)

For a node u of the gene tree, the node y of the species tree with largest rank such that δ(u) δ(y)

τ(y)

For a node y with rank i on the species tree, we denote τ(y) = τ i (the interval immediately above y)

λ i,j

The overall coalescence rate in interval τ i immediately preceding (backwards in time) the j th coalescence

H k

Number of sequences of coalescences above the root of the species tree starting with k lineages

f i

The joint density of coalescence times in interval τ i