  The Shapley value, a solution concept from cooperative game theory, has
recently been considered for both unrooted and rooted phylogenetic trees. Here,
we focus on the Shapley value of unrooted trees and first revisit the so-called
split counts of a phylogenetic tree and the Shapley transformation matrix that
allows for the calculation of the Shapley value from the edge lengths of a
tree. We show that non-isomorphic trees may have permutation-equivalent Shapley
transformation matrices and permutation-equivalent null spaces. This implies
that estimating the split counts associated with a tree or the Shapley values
of its leaves does not suffice to reconstruct the correct tree topology. We
then turn to the use of the Shapley value as a prioritization criterion in
biodiversity conservation and compare it to a greedy solution concept. Here, we
show that for certain phylogenetic trees, the Shapley value may fail as a
prioritization criterion, meaning that the diversity spanned by the top $k$
species (ranked by their Shapley values) cannot approximate the total diversity
of all $n$ species.
