  A Yule tree is the result of a branching process with constant birth and
death rates. Such a process serves as an instructive null model of many
empirical systems, for instance, the evolution of species leading to a
phylogenetic tree. However, often in phylogeny the only available information
is the pairwise distances between a small fraction of extant species
representing the leaves of the tree. In this article we study statistical
properties of the pairwise distances in a Yule tree. Using a method based on a
recursion, we derive an exact, analytic and compact formula for the expected
number of pairs separated by a certain time distance. This number turns out to
follow a increasing exponential function. This property of a Yule tree can
serve as a simple test for empirical data to be well described by a Yule
process. We further use this recursive method to calculate the expected number
of the $n$-most closely related pairs of leaves and the number of cherries
separated by a certain time distance. To make our results more useful for
realistic scenarios, we explicitly take into account that the leaves of a tree
may be incompletely sampled and derive a criterion for poorly sampled
phylogenies. We show that our result can account for empirical data, using two
families of birds species.
