This paper studies a common puzzle in statistical phylogenetics. When many estimated gene trees are averaged in the Billera–Holmes–Vogtmann (BHV) treespace, the resulting Fréchet mean tree sometimes has an internal node with three or more children (a multifurcation). The authors ask whether such a multifurcation reflects a true unresolved split in the species history (a hard polytomy) or is just sampling variability that would resolve with more data (a soft polytomy). They link this question to a geometric phenomenon called “stickiness,” where the sample Fréchet mean gets permanently trapped in a lower‑dimensional region of the space.

The main mathematical insight is that the Fréchet mean process hides a multidimensional random walk. By applying simple “folding” maps to the space, the authors show that each coordinate of this embedded process behaves like an independent random walk and that the time at which the mean becomes sticky is exactly determined by the largest last‑passage time above zero among these coordinates. Using this representation they obtain precise large‑sample asymptotics for the stickiness time, including explicit constants, under standard technical conditions used in large‑deviation theory.

Building on those asymptotics, the paper gives a fully nonparametric procedure to estimate, from observed gene trees, the probability that a trifurcation in the sample Fréchet mean will eventually bifurcate if more observations are collected. The authors prove the estimator has vanishing relative error as sample size grows. They also present a bootstrap bias‑corrected version that performs better in practice, and they provide code so others can reproduce the calculations.

As an application, the authors revisit a real example: the Fréchet mean of gene trees for eutherian mammals shows a trifurcation among tree shrews, glires (rodents and relatives), and primates. Using their bootstrap bias‑corrected estimator they assess the probability that this observed trifurcation is a soft polytomy — that is, whether additional data are likely to resolve the branching order.