  Phylogenetic analysis of DNA or other data commonly gives rise to a
collection or sample of inferred evolutionary trees. Principal Components
Analysis (PCA) cannot be applied directly to collections of trees since the
space of evolutionary trees on a fixed set of taxa is not a vector space. This
paper describes a novel geometrical approach to PCA in tree-space that
constructs the first principal path in an analogous way to standard linear
Euclidean PCA. Given a data set of phylogenetic trees, a geodesic principal
path is sought that maximizes the variance of the data under a form of
projection onto the path. Due to the high dimensionality of tree-space and the
nonlinear nature of this problem, the computational complexity is potentially
very high, so approximate optimization algorithms are used to search for the
optimal path. Principal paths identified in this way reveal and quantify the
main sources of variation in the original collection of trees in terms of both
topology and branch lengths. The approach is illustrated by application to
simulated sets of trees and to a set of gene trees from metazoan (animal)
species.
