Functional Data Analysis for tree-structured data objects

The field of FDA has made a lot of progress on the statistical analysis of the variation in a population of curves. A particularly challenging extension of this set of ideas, is to populations of tree-structured objects. Deep challenges arise, which involve a marriage of ideas from statistics, geometry, and numerical analysis, because the space of trees is strongly non-Euclidean in nature. These challenges, together with some first approaches to addressing them, are illustrated using a real data example, where each data point is the tree of blood vessels in one person s brain.