  The active field of Functional Data Analysis (about understanding the
variation in a set of curves) has been recently extended to Object Oriented
Data Analysis, which considers populations of more general objects. A
particularly challenging extension of this set of ideas is to populations of
tree-structured objects. We develop an analog of Principal Component Analysis
for trees, based on the notion of tree-lines, and propose numerically fast
(linear time) algorithms to solve the resulting optimization problems. The
solutions we obtain are used in the analysis of a data set of 73 individuals,
where each data object is a tree of blood vessels in one person's brain.
