Description Usage Arguments Value Author(s) References Examples

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. Its 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)=∑ rQI(T(Q)) over Q in P_4*

The rooted quartet index is a balance index.

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`tree` |
A rooted tree in phylo format. |

`shapeVal` |
A vector of length 5 containing the shape values |

`rQuartetI`

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

Sophie Kersting

T. M. Coronado, A. Mir, F. Rosselló, 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.

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