  In this work we study the limit distribution of an appropriately normalized
cophenetic index of the pure-birth tree conditioned on $n$ contemporary tips.
We show that this normalized phylogenetic balance index is a submartingale that
converges almost surely and in L2. We link our work with studies on trees
without branch lengths and show that in this case the limit distribution is a
contraction-type distribution, similar to the Quicksort limit distribution. In
the continuous branch case we suggest approximations to the limit distribution.
We propose heuristic methods of simulating from these distributions and it may
be observed that these algorithms result in reasonable tails. Therefore, we
propose a way based on the quantiles of the derived distributions for
hypothesis testing, whether an observed phylogenetic tree is consistent with
the pure-birth process. Simulating a sample by the proposed heuristics is
rapid, while exact simulation (simulating the tree and then calculating the
index) is a time-consuming procedure. We conduct a power study to investigate
how well the cophenetic indices detect deviations from the Yule tree and apply
the methodology to empirical phylogenies.
