Single-Elimination Tournament Bracket Count

A single-elimination tournament has $N$ teams (where $N$ is a power of 2). In each round, teams are paired up, and the loser of each match is eliminated. This continues until one champion remains. A "bracket" specifies which teams play each other in the first round, and how the winners are matched up in subsequent rounds. Two brackets are different if any first-round matchup differs, or if the way winners feed into later rounds differs. How many distinct tournament brackets are possible for $N$ labeled teams?