areaPerPairI {treebalance}

Calculation of the area per pair index for rooted trees

Description

This function calculates the area per pair index APP(T) for a given rooted tree T. The tree must not necessarily be binary. APP(T) is defined as

APP(T)=\frac{2}{n\cdot(n-1)}\cdot\sum_{1\leq i<j\leq n} d_T(i,j)

in which n denotes the number of leaves in T, and d_T(i,j) denotes the number of edges on the path between the two leaves i and j. Note that APP(T) can also be computed from the Sackin index S(T) and the total cophenetic index TCI(T) of T as APP(T)=\frac{2}{n}\cdot S(T)-\frac{4}{n(n-1)}\cdot TCI(T) enabling efficient computation.

The area per pair index does not fulfill the definition of an (im)balance index given in "Tree balance indices: a comprehensive survey" (Fischer et al., 2023).

For details on the area per pair index, see also Chapter 24 in "Tree balance indices: a comprehensive survey" (https://doi.org/10.1007/978-3-031-39800-1_24).

Usage

areaPerPairI(tree)

Arguments

tree

A rooted tree in phylo format.

Value

areaPerPairI returns the area per pair index of the given tree.

Author(s)

Luise Kuehn

References

T. Araujo Lima, F. M. D. Marquitti, and M. A. M. de Aguiar. Measuring Tree Balance with Normalized Tree Area. arXiv e-prints, art. arXiv:2008.12867, 2020.

Examples

tree <- ape::read.tree(text="((((,),),(,)),(((,),),(,)));")
areaPerPairI(tree)