Monotonicity of MRCA depth under the beta-splitting parameter

Establish whether, for every tree size n, sample size k, and nonnegative depth threshold r, the probability that the depth D_{n,k} of the most recent common ancestor of k uniformly sampled leaves is at most r is monotone increasing as a function of the Aldous beta-splitting parameter beta.

Background

The paper studies D_{n,k}, the depth of the most recent common ancestor of a uniformly sampled k-subset of leaves in a tree generated by the Aldous beta-splitting model. Its results indicate that increasing beta produces more balanced tree shapes and correspondingly smaller MRCA depths. The authors identify as an open problem the formal verification that this qualitative relationship holds uniformly for every n, k, and threshold r.

References

We end by pointing out two open problems. First, a general observation concerning our results is that increasing $\beta$ leads to a higher probability that $D_{n,k}$ is small. Thus, it may be of interest to formally establish whether or not, for each value of $n, k$ and $r$, $(D_{n,k} \leq r)$ is a monotone increasing function of $\beta$.

— Predicting the depth of the most recent common ancestor of a random sample of $k$ species: the impact of phylogenetic tree shape  (2501.09270 - Fuchs et al., 16 Jan 2025) in Section 5, “Concluding comments”