totCophI

This function calculates the total cophenetic index $TCI(T)$ of a given
rooted tree $T$. The tree must not necessarily be binary. $TCI(T)$
is defined as $$TCI(T)=\sum_{1\leq i&lt;j\leq n} \delta(lca(i,j))=\sum_{u\in V_{in}(T)\setminus\{\rho\}} binom(n_u,2)$$ in which $\delta(lca(i,j))$ denotes the depth of the lowest
common ancestor of the two leaves $i$ and $j$ and $V_{in}(T)\setminus\{\rho\}$
denotes the set of all inner vertices exept the root and $n_u$ denotes the
number of descendant leaves of $u$. The second formula is useful for efficient
computation of $TCI(T)$. The total cophenetic index is an imbalance index. 
For $n=1$ the function returns $TCI(T)=0$. 
For details on the total cophenetic index, see 
also Chapter 8 in "Tree balance indices: a comprehensive survey" (https://doi.org/10.1007/978-3-031-39800-1_8).

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

totCophI function

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

Arguments

Author

This function calculates the total cophenetic index $TCI(T)$ of a given
rooted tree $T$. The tree must not necessarily be binary. $TCI(T)$
is defined as $$TCI(T)=\sum_{1\leq i&lt;j\leq n} \delta(lca(i,j))=\sum_{u\in V_{in}(T)\setminus\{\rho\}} binom(n_u,2)$$ in which $\delta(lca(i,j))$ denotes the depth of the lowest
common ancestor of the two leaves $i$ and $j$ and $V_{in}(T)\setminus\{\rho\}$
denotes the set of all inner vertices exept the root and $n_u$ denotes the
number of descendant leaves of $u$. The second formula is useful for efficient
computation of $TCI(T)$. The total cophenetic index is an imbalance index. 
For $n=1$ the function returns $TCI(T)=0$. 
For details on the total cophenetic index, see 
also Chapter 8 in "Tree balance indices: a comprehensive survey" (https://doi.org/10.1007/978-3-031-39800-1_8).

Calculation of the total cophenetic index for rooted trees — totCophI

<dl>

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

</dl>

totCophI: Calculation of the total cophenetic index for rooted trees

Description

Usage

Value

Arguments

Author

References

Examples