(UNTESTED)

sub tree_tuples
{
  my( $tree, @names ) = @_;
  my $name = shift @names;
  map
  {
    my $k = $_;
    my $v = $tree->{$k};
    ref $v
      ? map( +{ $name => $k, %$_ }, tree_tuples( $v, @names ) )
      : { $name => $k, $names[0] => $v }
  }
  keys %$tree
}


# now use it with the OP's $hash hashref

print "Count Section Item\n";
for (
  sort {
    $b->{'count'} <=> $a->{'count'}
  or
    $a->{'section'} cmp $b->{'section'}
  or
    $a->{'item'} cmp $b->{'item'}
  } tree_tuples( $hash, qw( section item count ) ) )
{
  printf "%5d %7s %7s\n", @{$_}{qw( count section item )};
}