I have found another solution actually :-)

$sth_get_branch_list = $dbh->prepare(q(
     SELECT branch FROM postgres_database WHERE tree = 'first_tree'
     AND branch ~ ?;
));

$sth_get_branch_list = $dbh->prepare
 
$sth_get_branch_list->execute('^'.$root.'\.\d+$');

Thanks for the help though :-)
Chris