Fair soccer championship

[I got this question from Sergei Vorobyov.]

The games played in the soccer world championship form a binary tree, where only the winner of each game moves up the tree (ignoring the initial games, where the teams are placed into groups of 4, 2 of which of which go onto play in the tree of games I just described). Assuming that the teams can be totally ordered in terms of how good they are, the winner of the championship will indeed be the best of all of the teams. However, the second best team does not necessarily get a second place in the championship. How many additional games need to be played in order to determine the second best team?