stairs2

This function calculates the stairs2 value $st2(T)$ for a given rooted
binary tree $T$. It is defined as the mean ratio between the leaf
numbers of the smaller and larger pending subtree over all inner vertices, more precisely
$$st2(T)=\frac{1}{n-1}\cdot\sum_{u \in V_{in}(T)} \frac{n_{u_a}}{n_{u_b}}$$
in which $V_{in}(T)$ denotes the set of all inner vertices
of $T$, and in which $n_{u_a}\geq n_{u_b}$ denote the number of leaves
in the two pending subtrees that are
rooted at the direct descendants of $u$. The stairs2 value is an imbalance index. 
Special cases: For $n=1$, the function returns $st2(T)=0$ and a warning. 
For details on the stairs2 value, see 
also Chapter 23 in "Tree balance indices: a comprehensive survey" (https://doi.org/10.1007/978-3-031-39800-1_23).

The aim of the 'R' package 'treebalance' is to provide functions for the computation of
a large variety of (im)balance indices for rooted trees. The package accompanies the book
''Tree balance indices: a comprehensive survey'' by M. Fischer, L. Herbst, S. Kersting,
L. Kuehn and K. Wicke (2023) <ISBN: 978-3-031-39799-8>, <doi:10.1007/978-3-031-39800-1>, which gives a precise definition for the terms 'balance index' and 'imbalance index' (Chapter 4)
and provides an overview of the terminology in this manual (Chapter 2). For further information
on (im)balance indices, see also Fischer et al. (2021) <https://treebalance.wordpress.com>.
Considering both established and new (im)balance indices, 'treebalance' provides (among
others) functions for calculating the following 18 established indices and index families: the
average leaf depth, the B1 and B2 index, the Colijn-Plazzotta rank, the normal, corrected,
quadratic and equal weights Colless index, the family of Colless-like indices, the family of
I-based indices, the Rogers J index, the Furnas rank, the rooted quartet index, the s-shape
statistic, the Sackin index, the symmetry nodes index, the total cophenetic index and the
variance of leaf depths. Additionally, we include 9 tree shape statistics that satisfy the
definition of an (im)balance index but have not been thoroughly analyzed in terms of tree
balance in the literature yet. These are: the total internal path length, the total path length,
the average vertex depth, the maximum width, the modified maximum difference in widths, the
maximum depth, the maximum width over maximum depth, the stairs1 and the stairs2 index.
As input, most functions of 'treebalance' require a rooted (phylogenetic) tree in 'phylo' format
(as introduced in 'ape' 1.9 in November 2006). 'phylo' is used to store (phylogenetic) trees
with no vertices of out-degree one. For further information on the format we kindly refer the
reader to E. Paradis (2012) <http://ape-package.ird.fr/misc/FormatTreeR_24Oct2012.pdf>.

Luise Kuehn

treebalance

Computation of Tree (Im)Balance Indices

Mareike Fischer

Lina Herbst

Sophie Kersting

Kristina Wicke

stairs2 function

<dl><dt>tree</dt>
<dd>A rooted binary tree in phylo format.</dd></dl>

Arguments

Author

This function calculates the stairs2 value $st2(T)$ for a given rooted
binary tree $T$. It is defined as the mean ratio between the leaf
numbers of the smaller and larger pending subtree over all inner vertices, more precisely
$$st2(T)=\frac{1}{n-1}\cdot\sum_{u \in V_{in}(T)} \frac{n_{u_a}}{n_{u_b}}$$
in which $V_{in}(T)$ denotes the set of all inner vertices
of $T$, and in which $n_{u_a}\geq n_{u_b}$ denote the number of leaves
in the two pending subtrees that are
rooted at the direct descendants of $u$. The stairs2 value is an imbalance index. 
Special cases: For $n=1$, the function returns $st2(T)=0$ and a warning. 
For details on the stairs2 value, see 
also Chapter 23 in "Tree balance indices: a comprehensive survey" (https://doi.org/10.1007/978-3-031-39800-1_23).

Calculation of the stairs2 value for rooted binary trees — stairs2

<dl>

<dt>tree</dt>
<dd>A rooted binary tree in phylo format.</dd>

</dl>

stairs2: Calculation of the stairs2 value for rooted binary trees

Description

Usage

Value

Arguments

Author

References

Examples