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).
rQuartetI(tree, shapeVal = c(0, 1, 2, 3, 4))rQuartetI returns the rooted quartet index of the given tree based on the chosen shape values (see description for details).
A rooted tree in phylo format.
A vector of length 5 containing the shape values \(q_0,...,q_4\). Default is \((q_0,q_1,q_2,q_3,q_4)=(0,1,2,3,4)\).
Sophie Kersting
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.
tree <- ape::read.tree(text="((((,),),(,)),(((,),),(,)));")
rQuartetI(tree)
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