TotalCopheneticIndex() calculates the total cophenetic index
(Mir et al. 2013) for any tree, a measure of its balance;
TCIContext() lists its possible values.
TotalCopheneticIndex(x) TCIContext(x) # S3 method for numeric TCIContext(x)
A tree of class
$edge property, or a list thereof.
TotalCopheneticIndex() returns an integer denoting the total cophenetic index.
TCIContext() returns a data frame detailing the maximum and minimum value
obtainable for the Total Cophenetic Index for rooted binary trees with the
number of leaves of the given tree, and the expected value under the Yule
and Uniform models.
The variance of the expected value is given under the Yule model, but cannot
be obtained by calculation for the Uniform model.
The Total Cophenetic Index is a measure of tree balance -- i.e. whether a (phylogenetic) tree comprises symmetric pairs of nodes, or has a pectinate 'caterpillar' shape. The index has a greater resolution power than Sackin's and Colless' indices, and can be applied to trees that are not perfectly resolved.
For a tree with n leaves, the Total Cophenetic Index can take values of
The minimum value is higher for a perfectly resolved (i.e. dichotomous) tree
(see Lemma 14 of Mir et al. 2013).
Formulae to calculate the expected values under the Yule and Uniform models
of evolution are given in Theorems 17 and 23.
Full details are provided by Mir et al. (2013).
Mir A, Rosselló F, Rotger LA (2013). “A new balance index for phylogenetic trees.” Mathematical Biosciences, 241(1), 125--136. doi: 10.1016/j.mbs.2012.10.005 .
# Balanced trees have the minimum index for a binary tree; # Pectinate trees the maximum: TCIContext(8) #> maximum minimum uniform.expected yule.expected yule.variance #> 1 56 16 38.8345 28.51429 90.52281 TotalCopheneticIndex(PectinateTree(8)) #>  56 TotalCopheneticIndex(BalancedTree(8)) #>  16 TotalCopheneticIndex(StarTree(8)) #>  0 # Examples from Mir et al. (2013): tree12 <- ape::read.tree(text='(1, (2, (3, (4, 5))));') #Fig. 4, tree 12 TotalCopheneticIndex(tree12) # 10 #>  10 tree8 <- ape::read.tree(text='((1, 2, 3, 4), 5);') #Fig. 4, tree 8 TotalCopheneticIndex(tree8) # 6 #>  6 TCIContext(tree8) #> maximum minimum uniform.expected yule.expected yule.variance #> 1 10 5 8.285714 7.166667 5.138889 TCIContext(5L) # Context for a tree with 5 leaves. #> maximum minimum uniform.expected yule.expected yule.variance #> 1 10 5 8.285714 7.166667 5.138889