  Object Oriented Data Analysis is a new area in statistics that studies
populations of general data objects. In this article we consider populations of
tree-structured objects as our focus of interest. We develop improved analysis
tools for data lying in a binary tree space analogous to classical Principal
Component Analysis methods in Euclidean space. Our extensions of PCA are
analogs of one dimensional subspaces that best fit the data. Previous work was
based on the notion of tree-lines.
  In this paper, a generalization of the previous tree-line notion is proposed:
k-tree-lines. Previously proposed tree-lines are k-tree-lines where k=1. New
sub-cases of k-tree-lines studied in this work are the 2-tree-lines and
tree-curves, which explain much more variation per principal component than
tree-lines. The optimal principal component tree-lines were computable in
linear time. Because 2-tree-lines and tree-curves are more complex, they are
computationally more expensive, but yield improved data analysis results.
  We provide a comparative study of all these methods on a motivating data set
consisting of brain vessel structures of 98 subjects.
