I don't know whether you cut-and-pasted or typed it in directly, but some changes are below:

# 'strict' is your friend!
use strict;

...

# be sure to end each statement with a semicolon, and declar
# all variables with 'my'
my $sport = 'basket ball';
my $year  = '1999';

# Egads, escaping quotes is so un-Perlish. DBI
# placeholders make life much easier
my $sql = qq/ 
   SELECT round, teamname, score 
     FROM trees
    WHERE sport = ? and year = ?
 ORDER BY round DESC
/;

# Turn on DBI error checking -- anything with an error
# will immediately trigger 'die'
$dbh->{RaiseError} = 1;

# Prepare and plug in the values for the placeholders
my $sth5 = $dbh->prepare( $sql );
$sth5->execute( $sport, $year );

my $rowcount = 0;
my @list_of_teamdata = ();

# It's probably easier to do this with hashes, but we'll 
# stick with your use of arrays. Fetching arrayrefs can
# be faster, however, so we'll use those
while ( my $row = $sth5->fetchrow_arrayref() ) {
  my $round    = $row->[0];
  my $teamname = $row->[1];          
  my $score    = $row->[2]; 

  # Using 'qw' as you had done isn't what you wanted  -- 
  # instead, just create an anonymous arrayref and stick
  # it in the list of results         
  push @list_of_teamdata, 
     [ $round, $teamname, $score ];
  $rowcount++;
}

# Now print out your results -- use 'printf' for nicer
# formatting :-)
foreach my $info ( @list_of_teamdata ) {
  print "round $info->[0] team $info->[1] score $info->[2]\n";
}
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There are more concise ways of doing this, but I wanted to change things just enough to work. We can get to efficiency and readability later :-)

Chris
M-x auto-bs-mode