Landscapes of trees are mappings of tree space that are contoured according to some optimality criterion – often, but not necessarily, a tree’s score under a phylogenetic reconstruction technique (Bastert, Rockmore, Stadler, & Tinhofer, 2002). Detecting “islands” or “terraces” of trees can illuminate the nature of the space of optimal trees and thus inform tree search strategy (Maddison, 1991; Sanderson, McMahon, & Steel, 2011).

For simplicity (and to avoid scoring trees against a dataset), this example uses a tree’s balance (measured using the total cophenetic index) as its score (Mir, Rosselló, & Rotger, 2013). We assume that mappings have already been shown to be adequate (Smith, 2022).

A landscape is most simply visualized by colouring each tree according to its score:

# Load required libraries
library("TreeTools", quietly = TRUE)
library("TreeDist")

# Generate a set of trees
trees <- as.phylo(as.TreeNumber(BalancedTree(16)) + 0:100 - 15, 16)

# Create a 2D mapping
distances <- ClusteringInfoDist(trees)
mapping <- cmdscale(distances, 2)

# Score trees according to their balance
scores <- TotalCopheneticIndex(trees)

# Normalize scores
scoreMax <- TCIContext(trees[[1]])[["maximum"]]
scoreMin <- TCIContext(trees[[1]])[["minimum"]]
scores <- scores - scoreMin
scores <- scores / (scoreMax - scoreMin)

# Generate colour palette
col <- colorRamp(c("orange", "blue"))(scores)
rgbCol <- rgb(col, maxColorValue = 255)

# Plot trees, coloured by their score
plot(
  mapping,
  asp = 1, # Preserve aspect ratio - do not distort distances
  ann = FALSE, axes = FALSE, # Don't label axes: dimensions are meaningless
  col = rgbCol, # Colour trees by score
  pch = 16 # Plotting character: Filled circle
)

# Add a legend
PlotTools::SpectrumLegend(
  "left",
  title = "Tree balance",
  palette = rgb(colorRamp(c("orange", "blue"))(0:100 / 100) / 255),
  legend = floor(seq(scoreMax, scoreMin, length.out = 6)),
  bty = "n"
)

A more sophisticated output can be produced using contours, interpolating between adjacent trees. This example uses a simple inverse distance weighting function for interpolation; more sophisticated techniques such as kriging or (in continuous tree spaces) the use of geodesics (Khodaei, Owen, & Beerli, 2022) can produce even better results.

# Use an inverse distance weighting to interpolate between measured points
Predict <- function (x, y) {
  Distance <- function (a, b) {
    apply(a, 2, function (pt) sqrt(colSums((pt - b) ^ 2)))
  }
  predXY <- rbind(x, y)
  dists <- Distance(t(mapping), predXY)
  invDist <- 1 / dists
  weightings <- invDist / rowSums(invDist)

  # Return:
  colSums(scores * t(weightings))
}

# Generate grid for contour plot
resolution <- 32
xLim <- range(mapping[, 1]) * 1.1
yLim <- range(mapping[, 2]) * 1.11
x <- seq(xLim[1], xLim[2], length.out = resolution)
y <- seq(yLim[1], yLim[2], length.out = resolution)
z <- outer(x, y, Predict) # Predicted values for each grid square

# Plot
filled.contour(
  x, y, z,
  asp = 1, # Preserve aspect ratio - do not distort distances
  ann = FALSE, axes = FALSE, # Don't label axes: dimensions are meaningless
  plot.axes = {points(mapping, xpd = NA)} # Use filled.contour coordinates
)

if (requireNamespace("plotly", quietly = TRUE)) {
  library("plotly", quietly = TRUE)
  fig <- plot_ly(x = x, y = y, z = z)
  fig <- fig %>% add_surface()
  fig
} else {
  print("Run `install.packages('plotly')` to view this output")
}

References

Bastert, O., Rockmore, D., Stadler, P. F., & Tinhofer, G. (2002). Landscapes on spaces of trees. Applied Mathematics and Computation, 131(2-3), 439–459. doi: 10.1016/S0096-3003(01)00164-3

Khodaei, M., Owen, M., & Beerli, P. (2022). Geodesics to characterize the phylogenetic landscape. bioR\(\chi\)iv. doi: 10.1101/2022.05.11.491507

Maddison, D. R. (1991). The discovery and importance of multiple islands of most-parsimonious trees. Systematic Biology, 40(3), 315–328. doi: 10.1093/sysbio/40.3.315

Mir, A., Rosselló, F., & Rotger, L. A. (2013). A new balance index for phylogenetic trees. Mathematical Biosciences, 241(1), 125–136. doi: 10.1016/j.mbs.2012.10.005

Sanderson, M. J., McMahon, M. M., & Steel, M. (2011). Terraces in Phylogenetic Tree Space. Science, 333(6041), 448–450. doi: 10.1126/science.1206357

Smith, M. R. (2022). Robust analysis of phylogenetic tree space. Systematic Biology, 71(5), 1255–1270. doi: 10.1093/sysbio/syab100

Analysing landscapes of phylogenetic trees

Martin R. Smith

References