Calculate the Kendall–Colijn tree distance, a measure related to the path difference.

KendallColijn(tree1, tree2 = NULL, Vector = KCVector)

KCVector(tree)

PathVector(tree)

SplitVector(tree)

KCDiameter(tree)

## Arguments

tree1, tree2

Trees of class phylo, with leaves labelled identically, or lists of such trees to undergo pairwise comparison. Where implemented, tree2 = NULL will compute distances between each pair of trees in the list tree1 using a fast algorithm based on Day (1985).

Vector

Function converting a tree to a numeric vector.

KCVector, the default, returns the number of edges between the common ancestor of each pair of leaves and the root of the tree (per Kendall and Colijn 2016) .

PathVector returns the number of edges between each pair of leaves (per Steel and Penny 1993) .

SplitVector returns the size of the smallest split that contains each pair of leaves (per Smith 2022 ).

tree

A tree of class phylo.

## Value

KendallColijn() returns an array of numerics providing the distances between each pair of trees in tree1 and tree2, or splits1 and splits2.

KCDiameter() returns the value of the Kendall & Colijn's (2016) metric distance between two pectinate trees with n leaves ordered in the opposite direction, which I suggest (without any attempt at a proof) may be a useful proxy for the diameter (i.e. maximum value) of the K–C metric.

## Details

The Kendall–Colijn distance works by measuring, for each pair of leaves, the distance from the most recent common ancestor of those leaves and the root node. For a given tree, this produces a vector of values recording the distance-from-the-root of each most recent common ancestor of each pair of leaves.

Two trees are compared by taking the Euclidean distance between the respective vectors. This is calculated by taking the square root of the sum of the squares of the differences between the vectors.

An analogous distance can be created from any vector representation of a tree. The split size vector metric (Smith 2022) is an attempt to mimic the Kendall Colijn metric in situations where the position of the root should not be afforded special significance; and the path distance (Steel and Penny 1993) is a familiar alternative whose underlying vector measures the distance of the last common ancestor of each pair of leaves from the leaves themselves, i.e. the length of the path from one leaf to another.

None of these vector-based methods performs as well as other tree distances in measuring similarities in the relationships implied by a pair of trees (Smith 2020) ; in particular, the Kendall Colijn metric is strongly influenced by tree balance, and may not be appropriate for a suite of common applications (Smith 2022) .

## Functions

• KCVector(): Creates a vector that characterises a rooted tree, as described in Kendall and Colijn (2016) .

• PathVector(): Creates a vector reporting the number of edges between each pair of leaves, per the path metric of Steel and Penny (1993) .

• SplitVector(): Creates a vector reporting the smallest split containing each pair of leaves, per the metric proposed in Smith (2022) .

treespace::treeDist is a more sophisticated, if more cumbersome, implementation that supports lambda > 0, i.e. use of edge lengths in tree comparison.

Other tree distances: JaccardRobinsonFoulds(), MASTSize(), MatchingSplitDistance(), NNIDist(), NyeSimilarity(), PathDist(), Robinson-Foulds, SPRDist(), TreeDistance()

## Examples

KendallColijn(TreeTools::BalancedTree(8), TreeTools::PectinateTree(8))
#>  11.48913

set.seed(0)
KendallColijn(TreeTools::BalancedTree(8), lapply(rep(8, 3), ape::rtree))
#>  9.591663 5.567764 9.949874
KendallColijn(lapply(rep(8, 4), ape::rtree))
#>          1        2        3
#> 2 7.280110
#> 3 7.874008 8.185353
#> 4 4.795832 7.071068 7.681146

KendallColijn(lapply(rep(8, 4), ape::rtree), Vector = SplitVector)
#>           1         2         3
#> 2 10.862780
#> 3 10.148892 12.124356
#> 4  8.000000 11.489125  8.185353

# Notice that changing tree shape close to the root results in much
# larger differences
tree1 <- ape::read.tree(text = "(a, (b, (c, (d, (e, (f, (g, h)))))));")
tree2 <- ape::read.tree(text = "(a, ((b, c), (d, (e, (f, (g, h))))));")
tree3 <- ape::read.tree(text = "(a, (b, (c, (d, (e, ((f, g), h))))));")
trees <- c(tree1, tree2, tree3)
KendallColijn(trees)
#>          1        2
#> 2 4.000000
#> 3 1.414214 4.242641
KendallColijn(trees, Vector = SplitVector)
#>          1        2
#> 2 2.449490
#> 3 2.449490 3.162278
KCDiameter(trees)
#> Warning: first element used of 'length.out' argument
#> Warning: longer object length is not a multiple of shorter object length
#>  15.87451
KCDiameter(4)
#>  3.162278