  A taxonomy is a standardized framework to classify and organize items into
categories. Hierarchical taxonomies are ubiquitous, ranging from the
classification of organisms to the file system on a computer. Characterizing
the typical distribution of items within taxonomic categories is an important
question with applications in many disciplines. Ecologists have long sought to
account for the patterns observed in species-abundance distributions (the
number of individuals per species found in some sample), and computer
scientists study the distribution of files per directory. Is there a universal
statistical distribution describing how many items are typically found in each
category in large taxonomies? Here, we analyze a wide array of large,
real-world datasets -- including items lost and found on the New York City
transit system, library books, and a bacterial microbiome -- and discover such
an underlying commonality. A simple, non-parametric branching model that
randomly categorizes items and takes as input only the total number of items
and the total number of categories successfully reproduces the abundance
distributions in these datasets. This result may shed light on patterns in
species-abundance distributions long observed in ecology. The model also
predicts the number of taxonomic categories that remain unrepresented in a
finite sample.
