Split-selection methods

The split-selection methods differ only in which groupings they keep, so they form a nested sequence of increasing resolution. To see this, take seven trees that share a backbone but disagree on the placement of a few leaves:

trees <- ape::read.tree(text = c(
  "((((((t1,t3),t2),t4),(t5,t6)),t7),(t8,t9));",
  "((((((t1,t3),(t5,t6)),t4),t2),t7),(t8,t9));",
  "((((((t1,t2),t3),t4),(t5,t6)),t7),(t8,t9));",
  "(((((t1,(t2,t3)),(t5,t6)),t4),t7),(t8,t9));",
  "((((((t1,t2),t3),t4),(t7,(t5,t6))),t9),t8);",
  "((((((t1,t3),t2),(t5,t6)),t4),t7),(t8,t9));",
  "((((t1,((t2,t3),(t5,t6))),t4),(t8,t9)),t7);"))

# Colour leaf labels consistently
nTip <- 9
leafCol <- setNames(hcl.colors(nTip + 1), TipLabels(nTip + 1))
plotCons <- function(tree, main = "") {
  plot(tree, tip.color = leafCol[tree$tip.label], main = main,
       font = 2, cex = 1, edge.width = 1.5)
}

oldPar <- par(mfrow = c(2, 4), mar = c(0.5, 0.5, 1.5, 0.5))
for (i in seq_along(trees)) plotCons(trees[[i]], main = paste("Tree", i))
par(oldPar)

Each method retains a superset of the groupings kept by the one before it:

oldPar <- par(mfrow = c(2, 2), mar = c(0.5, 0.5, 1.5, 0.5))
plotCons(Strict(trees),    "Strict")
plotCons(Majority(trees),  "Majority-rule")
plotCons(Frequency(trees), "Frequency difference")
plotCons(Greedy(trees),    "Greedy (extended majority)")

par(oldPar)

data.frame(
  method  = c("Strict", "Majority", "Frequency", "Greedy"),
  splits  = c(NSplits(Strict(trees)),    NSplits(Majority(trees)),
              NSplits(Frequency(trees)),  NSplits(Greedy(trees)))
)
#>      method splits
#> 1    Strict      2
#> 2  Majority      4
#> 3 Frequency      5
#> 4    Greedy      6

Strict() keeps only the two groupings that occur in every tree; Majority() adds those that occur in more than half; Frequency() keeps any grouping that beats every grouping it conflicts with; and Greedy() adds compatible groupings, most frequent first, giving the most resolved summary.

Simply counting splits overlooks the possibility that some splits are more heavily contradicted by other trees. Loose() (the semi-strict, or combinable-component, consensus) keeps every grouping that no tree contradicts, whereas MajorityPlus() keeps any grouping that is displayed by more trees than contradict it.

Rooted methods

Adams(), RStar() and Local() treat the input as rooted and reason about clusters and rooted triplets rather than unrooted splits. They may therefore recover structure that an unrooted strict consensus would collapse:

rTrees <- ape::read.tree(text = c(
  "((((t1,t2),t3),(((t5,t6),t4),t7)),t8);",
  "(((((t1,t2),t3),(t5,(t6,t4))),t7),t8);",
  "(((((t1,t2),t3),((t5,t4),t6)),t7),t8);",
  "((((((t1,t2),t4),(t5,t6)),t3),t7),t8);",
  "((((((t1,t2),t3),t4),(t5,t6)),t7),t8);",
  "(((((t1,t2),t4),(t3,(t5,t6))),t7),t8);"))

oldPar <- par(mfrow = c(2, 3), mar = c(0.5, 0.5, 1.5, 0.5))
for (i in seq_along(rTrees)) plotCons(rTrees[[i]], main = paste("Tree", i))

par(oldPar)

The unrooted strict consensus keeps a single grouping, but the rooted methods recover more:

oldPar <- par(mfrow = c(2, 2), mar = c(0.5, 0.5, 1.5, 0.5))
plotCons(Strict(rTrees), "Strict")
plotCons(Adams(rTrees),  "Adams")
plotCons(RStar(rTrees),  "RStar")
plotCons(Local(rTrees),  "Local")

par(oldPar)

data.frame(
  method = c("Strict", "Adams", "RStar", "Local"),
  splits = c(NSplits(Strict(rTrees)), NSplits(Adams(rTrees)),
             NSplits(RStar(rTrees)),  NSplits(Local(rTrees)))
)
#>   method splits
#> 1 Strict      1
#> 2  Adams      4
#> 3  RStar      4
#> 4  Local      3

An Adams() consensus may display groupings that no input tree contains. RStar() keeps each rooted triplet that wins a plurality over both alternatives; Local() returns the minimum local consensus of the shared triplets (limited to 20 leaves, and best suited to congruent samples).

Distance and branch-length summaries

Maximizing the counts of individual groupings is equivalent to finding the tree with the minimum total Robinson–Foulds distance to all input trees. As the Robinson–Foulds distance exhibits shortcomings (Smith, 2020), other distance measures potentially provide more instructive summary trees.

Quartet() is the quartet-distance analogue, seeking (an approximation to) the tree that minimizes the total quartet distance to the inputs instead (Takazawa et al., 2026). Because the quartet distance gives extra weight to deep branches, the result is often more resolved than the majority-rule tree:

c(majority = NSplits(Majority(trees)),
  quartet  = NSplits(Quartet(trees)))
#> majority  quartet 
#>        4        5

median.multiPhylo() in the ‘TreeDist’ package allows such a median to be selected from within a sample of trees under any metric; unlike the previous approaches, this is restricted to tree topologies within the sample.

When the trees carry edge lengths, two further summaries become available. Average() returns the tree whose path-length (patristic) distances best match the average of the input distance matrices, while BHVMean() computes the Fréchet mean in Billera–Holmes–Vogtmann treespace, with branch lengths; BHVDistance(), BHVPairwiseDistances() and BHVVariance() provide the underlying geodesic distances and dispersion.

blTrees <- ape::read.tree(text = c(
  "(((t1:0.64,t2:0.84):0.52,(t3:0.84,t4:0.87):0.42):0.46,(t5:0.68,t6:0.34):0.70,t7:0.42);",
  "(((t1:0.64,t2:0.40):0.72,(t3:0.87,t4:0.38):0.87):0.41,(t5:0.58,t6:0.63):0.80,t7:0.63);",
  "(((t1:0.53,t2:0.50):0.78,(t3:0.66,t4:0.37):0.66):0.40,(t5:0.65,t6:0.68):0.48,t7:0.61);",
  "(((t1:0.48,t2:0.47):0.31,(t3:0.46,t4:0.73):0.79):0.65,(t5:0.87,t6:0.34):0.84,t7:0.75);",
  "(((t1:0.85,t2:0.47):0.71,(t4:0.72,(t5:0.78,t6:0.87):0.62):0.36):0.42,t3:0.37,t7:0.46);"))
meanTree <- BHVMean(blTrees)
oldPar <- par(mfrow = c(2, 3), mar = c(0.5, 0.5, 1.5, 0.5))
for (i in seq_along(blTrees)) plotCons(blTrees[[i]], main = paste("Tree", i))
plotCons(meanTree, main = "BHV mean")

par(oldPar)

BHVVariance(blTrees, mean = meanTree)
#> [1] 0.5749739

The mean recovers the shared topology. The two groupings that every tree displays keep their full length; the two that the fifth tree contradicts exhibit shorter branches.

Method selection

The most appropriate method in a particular circumstance depends on the value of precision versus accuracy; trees that resolve more groupings are progressively less likely to be accurate (Smith, 2019).

By omitting unstable leaves from the input trees, perhaps via the ‘Rogue’ package, it is possible to improve the resolution of consensus trees by reconciling groupings that only differ in the position of rogue taxa (Smith, 2022).