This function calculates the B2 index \(B2(T)\) for a given rooted
tree \(T\). The tree must not necessarily be binary. \(B2(T)\) is defined as
$$B2(T)=-\sum_{x\in V_L(T)} p_x\cdot log(p_x)$$ in which \(V_L(T)\) denotes the leaf
set of \(T\), and in which $$p_x=\prod_{v\in anc(x)} \frac{1}{|child(v)|}$$ denotes
the probability of reaching leaf \(x\) when starting at the root and assuming
equiprobable branching at each vertex \(v\in anc(x)\) with \(anc(x)\)
denoting the set of ancestors of \(x\) excluding
\(x\). \(child(v)\) denotes the set of children of the inner vertex \(v\).
The \(B2\) index is a balance index.
For \(n=1\) the function returns \(B2(T)=0\) and a warning.
For details on the B2 index, see
also Chapter 11 in "Tree balance indices: a comprehensive survey" (https://doi.org/10.1007/978-3-031-39800-1_11).
B2I(tree, logbase = 2)B2I returns the B2 index of the given tree.
A rooted tree in phylo format.
The base that shall be used for the logarithm. For binary trees it is common to use base 2.
Sophie Kersting, Luise Kuehn
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P.-M. Agapow and A. Purvis. Power of Eight Tree Shape Statistics to Detect Nonrandom Diversification: A Comparison by Simulation of Two Models of Cladogenesis. Systematic Biology, 51(6):866-872, 2002.doi: 10.1080/10635150290102564.
URL https://doi.org/10.1080/10635150290102564.
M. Hayati, B. Shadgar, and L. Chindelevitch. A new resolution function to evaluate tree shape statistics. PLOS ONE, 14(11):e0224197, 2019. doi: 10.1371/journal.pone.0224197.
URL https://doi.org/10.1371/journal.pone.0224197.
M. Kirkpatrick and M. Slatkin. Searching for evolutionary patterns in the shape of a phylogenetic tree. Evolution, 47(4):1171-1181, 1993. doi: 10.1111/j.1558-5646.1993.tb02144.x.
tree <- ape::read.tree(text="((((,),),(,)),(((,),),(,)));")
B2I(tree)
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