$where = array(
  $this->left_column_name.' <' => $left,
  $this->right_column_name.' >' => $rgt,
);
$this->db->limit(1);

//아래 줄을 추가하였습니다

$this->db->order_by($this->right_column_name, 'asc'); 

$query = $this->db->get_where($this->table_name, $where);
if ($query->num_rows() > 0) {
  return $query->row();
}
return FALSE;

CREATE TABLE IF NOT EXISTS `board_category` (
  `id` int(11) NOT NULL AUTO_INCREMENT,
  `name` varchar(50) NOT NULL,
  `controller_name` varchar(50) DEFAULT NULL,
  `url` varchar(50) DEFAULT NULL,
  `lft` int(11) NOT NULL DEFAULT '0',
  `rgt` int(11) NOT NULL DEFAULT '0',
  `parent_id` int(11) NOT NULL DEFAULT '0',
  PRIMARY KEY (`id`),
  KEY `lft` (`lft`),
  KEY `rgt` (`rgt`),
  KEY `parent_id` (`parent_id`)
);