  In order to develop statistical methods for shapes with a tree-structure, we
construct a shape space framework for tree-like shapes and study metrics on the
shape space. This shape space has singularities, corresponding to topological
transitions in the represented trees. We study two closely related metrics on
the shape space, TED and QED. QED is a quotient Euclidean distance arising
naturally from the shape space formulation, while TED is the classical tree
edit distance. Using Gromov's metric geometry we gain new insight into the
geometries defined by TED and QED. We show that the new metric QED has nice
geometric properties which facilitate statistical analysis, such as existence
and local uniqueness of geodesics and averages. TED, on the other hand, does
not share the geometric advantages of QED, but has nice algorithmic properties.
We provide a theoretical framework and experimental results on synthetic data
trees as well as airway trees from pulmonary CT scans. This way, we effectively
illustrate that our framework has both the theoretical and qualitative
properties necessary to build a theory of statistical tree-shape analysis.
