Uniqueness of Fréchet means of extension spaces

Determine whether the Fréchet mean of BHV extension spaces E_{T_1}^{𝒩}, E_{T_2}^{𝒩}, …, E_{T_r}^{𝒩} for collections of phylogenetic trees with non-identical leaf sets is unique in applied data examples.

Background

The paper introduces distances between extension spaces in BHV tree space to compare phylogenetic trees with non-identical leaf sets. Building on this, the authors suggest a "supertree" approach via Fréchet means of extension spaces for collections of trees, analogous to means in standard BHV space.

While Fréchet means in the standard BHV space are unique, minimum-distance paths between extension spaces need not be unique, raising the possibility that Fréchet means of extension spaces may also be non-unique. In their applied example, the authors observed two minimal paths with the same midpoint, motivating a conjecture that uniqueness may often hold in practice and indicating further work to formalize and paper these means.