A Mathematical Connection Between Single-Elimination Sports Tournaments and Evolutionary Trees

Abstract

How many ways are there to arrange the sequence of games in a single-elimination sports tournament? We consider the connection between this enumeration problem and the enumeration of “labeled histories,” or sequences of asynchronous branching events, in mathematical phylogenetics. The possibility of playing multiple games simultaneously in different arenas suggests an extension of the enumeration of labeled histories to scenarios in which multiple branching events occur simultaneously. We provide a recursive result enumerating game sequences and labeled histories in which simultaneity is allowed. For a March Madness basketball tournament of 68 labeled teams, the number of possible sequences of games is ∼1.91×1078 if arbitrarily many arenas are available, but only ∼3.60×1068 if all games must be played sequentially in the same arena.