rQuartetI {treebalance}

Calculation of the rooted quartet index for rooted trees

Description

This function calculates the rooted quartet index rQI(T) for a given rooted tree T. The tree must not necessarily be binary.

Let T be a rooted tree, whose leaves are 1,...,n. Let P_4 denote the set of all subsets of \{1,...,n\} that have cardinality 4. Let T(Q) denote the rooted quartet on Q\in P_4 that is obtained by taking the subgraph of T that is induced by Q and supressing its outdegree-1 vertices. T(Q) can have one of the five following shapes:

- Q_0^*: This is the caterpillar tree shape on 4 leaves, i.e. "(,(,(,)));" in Newick format. It has 2 automorphisms.
- Q_1^*: This is the tree shape on 4 leaves that has three pending subtrees rooted at the children of the root of T, one of them being a cherry and the other two being single vertices, i.e. "((,),,);" in Newick format. It has 4 automorphisms.
- Q_2^*: This is the tree shape on 4 leaves that has two pending subtrees rooted at the children of the root of T, one of them being a star tree shape on 3 leaves and the other one being a single vertex, i.e. "((,,),);" in Newick format. It has 6 automorphisms.
- Q_3^*: This is the fully balanced binary tree shape on 4 leaves, i.e. "((,),(,));" in Newick format. It has 8 automorphisms.
- Q_4^*: This is the star tree shape on 4 leaves, i.e. "(,,,);" in Newick format. It has 24 automorphisms.

T(Q) is assigned an rQI-value based on its shape, i.e. rQI(T(Q))=q_i if T(Q) has the shape Q_i^*. The values q_0,...,q_4 are chosen in such a way that they increase with the symmetry of the shape as measured by means of its number of automorphisms. Coronado et al. (2019) suggested the values q_0=0 and q_i=i or q_i=2^i for i=1,...,4.
The rooted quartet index rQI(T) of the tree T is then defined as the sum of the rQI-values of its rooted quartets:

rQI(T)=\sum_{Q\in P_4} rQI(T(Q))

The rooted quartet index is a balance index.

For details on the rooted quartet index, see also Chapter 20 in "Tree balance indices: a comprehensive survey" (https://doi.org/10.1007/978-3-031-39800-1_20).

Usage

rQuartetI(tree, shapeVal = c(0, 1, 2, 3, 4))

Arguments

tree

A rooted tree in phylo format.

shapeVal

A vector of length 5 containing the shape values q_0,...,q_4. Default is (q_0,q_1,q_2,q_3,q_4)=(0,1,2,3,4).

Value

rQuartetI returns the rooted quartet index of the given tree based on the chosen shape values (see description for details).

Author(s)

Sophie Kersting

References

T. M. Coronado, A. Mir, F. Rossello, and G. Valiente. A balance index for phylogenetic trees based on rooted quartets. Journal of Mathematical Biology, 79(3):1105-1148, 2019. doi: 10.1007/s00285-019-01377-w. URL https://doi.org/10.1007/s00285-019-01377-w.

Examples

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