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Complex data objects arise in many areas of modern science includingevolutionary biology, nueroscience, dynamics of gene expression and medicalimaging. Object oriented data analysis (OODA) is the statistical analysis ofdatasets of complex objects. Data analysis of tree data objects is an excitingresearch area with interesting questions and challenging problems. This thesisfocuses on tree oriented statistical methodologies, and algorithms for solvingrelated mathematical optimization problems.  This research is motivated by the goal of analyzing a data set of images ofhuman brain arteries. The approach we take here is to use a novelrepresentation of brain artery systems as points in phylogenetic treespace. Thetreespace property of unique global geodesics leads to a notion of geometriccenter called a Fr\'echet mean. For a sample of data points, the Fr\'echetfunction is the sum of squared distances from a point to the data points, andthe Fr\'echet mean is the minimizer of the Fr\'echet function.  In this thesis we use properties of the Fr\'echet function to develop analgorithmic system for computing Fr\'echet means. Properties of the Fr\'echetfunction are also used to show a sticky law of large numbers which describes asurprising stability of the topological tree structure of sample Fr\'echetmeans at that of the population Fr\'echet mean. We also introducenon-parametric regression of brain artery tree structure as a response variableto age based on weighted Fr\'echet means.

Tree Oriented Data Analysis