This is yet another fix for the problem of upgrading Wordpress on the Byet hosting free accounts.

I have Wordpress installed on the root of my domain. This is a combination of several procedures that I have found that did not work individually for me, but when combined did upgrade Wordpress.

1.Use the control panel to open the file manager and edit your Wordpress wp-config.php by entering these lines at the end of the file:

Code:
/* Fix Wordpress autoupgrades on Byethost.com */

putenv('TMPDIR=' . ini_get('upload_tmp_dir'));
if(is_admin()) {
add_filter('filesystem_method', create_function('$a', 'return "direct";' ));
define( 'FS_CHMOD_DIR', 0751 );
}
2.When executing the upgrade within Wordpress; I needed to use the following data:

HOSTNAME: your ftp hostname(from your control panel)/your ftpusername(from your control panel)/your domain/htdocs
Be sure to replace the above standin values with your actual data from your control panel and also ensure that you include your entire path to Wordpress if yours is not installed in the root of your domain.

USERNAME: your ftp username(from your control panel)

PASSWORD: (your hosting account password)

This worked for me and prevented the "can't find Wordpress root directory" error. I hope it solves your issues also.
Best Regards,
Dan Pueppke