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)
```

- 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 & Colijn 2016).`PathVector`

returns the number of edges between each pair of leaves (per Steel & 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`

.

`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.

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 Euclidian 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.

This metric emphasizes the position of the root; the path difference instead 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.

`KCVector`

: Creates a vector that characterises a rooted tree, as described in Kendall & Colijn (2016).`PathVector`

: Creates a vector reporting the path length between each pair of leaves, per the path metric of Steel & Penny (1993).`SplitVector`

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

Kendall M, Colijn C (2016).
“Mapping phylogenetic trees to reveal distinct patterns of evolution.”
*Molecular Biology and Evolution*, **33**(10), 2735--2743.
doi:10.1093/molbev/msw124
.

Smith MR (2022).
“Robust analysis of phylogenetic tree space.”
*Systematic Biology*, syab100.
doi:10.1093/sysbio/syab100
.

`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()`

```
KendallColijn(TreeTools::BalancedTree(8), TreeTools::PectinateTree(8))
#> [1] 11.48913
set.seed(0)
KendallColijn(TreeTools::BalancedTree(8), lapply(rep(8, 3), ape::rtree))
#> [1] 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
#> [1] 15.87451
KCDiameter(4)
#> [1] 3.162278
```