A plays B.  call the winner A by symmetry.
A plays C
IF A WINS, 
   D plays E.  call the winner D by symmetry.  A plays D twice
      IF A WINS BOTH, A plays E 3 times and E needs to win all 3
      IF THEY SPLIT, A plays D best-of-3
      IF D WINS BOTH, A plays B and winner plays D twice (D wins ties)
IF C WINS, A and C play again
   IF A WINS, A plays D
      IF A WINS, C plays E and the winner plays A best-of-3
      IF D WINS, C plays D
         IF C WINS, A plays D, the winner plays E, and the winner plays C
         IF D WINS, A plays D, the winner plays E, and the winner plays C
   IF C WINS, C plays D
      IF C WINS, C plays E
         IF C WINS, B plays C 3 times and needs to win all 3
         IF E WINS, C plays E best-of-3
      IF D WINS, C plays D twice more
         IF C WINS BOTH, C plays E twice (C wins ties)
         IF THEY SPLIT, C plays E and the winner plays D
         IF D WINS BOTH, D plays E twice (D wins ties)