  The leaves of angiosperms contain highly complex venation networks consisting
of recursively nested, hierarchically organized loops. We describe a new
phenotypic trait of reticulate vascular networks based on the topology of the
nested loops. This phenotypic trait encodes information orthogonal to widely
used geometric phenotypic traits, and thus constitutes a new dimension in the
leaf venation phenotypic space. We apply our metric to a database of 186 leaves
and leaflets representing 137 species, predominantly from the Burseraceae
family, revealing diverse topological network traits even within this single
family. We show that topological information significantly improves
identification of leaves from fragments by calculating a "leaf venation
fingerprint" from topology and geometry. Further, we present a phenomenological
model suggesting that the topological traits can be explained by noise effects
unique to specimen during development of each leaf which leave their imprint on
the final network. This work opens the path to new quantitative identification
techniques for leaves which go beyond simple geometric traits such as vein
density and is directly applicable to other planar or sub-planar networks such
as blood vessels in the brain.
