totCophI {treebalance}

Calculation of the total cophenetic index for rooted trees

Description

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<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).

Usage

totCophI(tree)

Arguments

tree

A rooted tree in phylo format.

Value

totCophI returns the total cophenetic index of the given tree.

Author(s)

Sophie Kersting

References

A. Mir, F. Rossello, and L. Rotger. A new balance index for phylogenetic trees. Mathematical Bio-sciences, 241(1):125-136, 2013. doi: 10.1016/j.mbs.2012.10.005.

Examples

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