  In this paper we prove a combinatorial theorem for finite labellings of
trees, and show that it is equivalent to a theorem for finite covers of metric
trees and a fixed point theorem on metric trees. We trace how these connections
mimic the equivalence of the Brouwer fixed point theorem with the classical KKM
lemma and Sperner's lemma. We also draw connections to a KKM-type theorem about
infinite covers of metric trees and fixed point theorems for non-compact metric
trees. Finally, we develop a new KKM-type theorem for cycles, and discuss
interesting social consequences, including an application in voting theory.
