$\mathcal{T}$

Species tree with realvalued divergence times

$\mathcal{G}$

Ranked gene tree (realvalued coalescence times not specified)

n

The number of leaves of $\mathcal{T}$ and $\mathcal{G}$

s
_{
i
}

Speciation times, with s_{1} >⋯> s_{n−1}, let s_{0} = ∞

τ
_{
i
}

Intervals between speciation times, τ_{
i
}= [s_{
i
},s_{i−1})

ℓ
_{
i
}

The number of gene tree lineages at time s_{
i
}

m
_{
i
}

The number of coalescence events in interval τ_{
i
}

${\mathcal{G}}_{i,{\ell}_{i}}$

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, u_{1} is most ancient, u_{n−1} is the most recent

k
_{i,j,z}

The number of lineages available for coalescence in population y_{i,z} just after the j th coalescence (considered forward in time) in interval τ_{
i
}; k_{i,0,z} is the number of lineages “exiting” at time s_{i−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
}
