How to generate a tree view from this result set based on Tree Traversal Algorithm?

I have this table:

CREATE TABLE `categories` (
  `id` int(11) NOT NULL auto_increment,
  `category_id` int(11) default NULL,
  `root_id` int(11) default NULL,
  `name` varchar(100) collate utf8_unicode_ci NOT NULL,
  `lft` int(11) NOT NULL,
  `rht` int(11) NOT NULL,
  PRIMARY KEY  (`id`),
  KEY `category_id` (`category_id`),
  KEY `lft` (`lft`,`rht`),
  KEY `root_id` (`root_id`)
) 

Based on this question: Getting a modified preorder tree traversal model (nested set) into a <ul>

The difference is that I have many trees in one table. Each row has a foreign key representing its parent and its top parent: category_id and root_id. Also I have the lft and rht fields based on this example: http://articles.sitepoint.com/article/hierarchical-data-database/2

Based on this rows:

INSERT INTO `categories` VALUES(1, NULL, NULL, 'Fruits', 1, 14);
INSERT INTO `categories` VALUES(2, 1, 1, 'Apple', 2, 3);
INSERT INTO `categories` VALUES(3, 1, 1, 'Orange', 4, 9);
INSERT INTO `categories` VALUES(4, 3, 1, 'Orange Type 1', 5, 6);
INSERT INTO `categories` VALUES(5, 3, 1, 'Orange Type 2', 7, 8);
INSERT INTO `categories` VALUES(6, 1, 1, 'Pear', 10, 11);
INSERT INTO `categories` VALUES(7, 1, 1, 'Banana', 12, 13);
INSERT INTO `categories` VALUES(8, NULL, NULL, 'Eletronics', 1, 14);
INSERT INTO `categories` VALUES(9, 8, 8, 'Cell Phones', 2, 3);
INSERT INTO `categories` VALUES(10, 8, 8, 'Computers', 4, 9);
INSERT INTO `categories` VALUES(11, 10, 8, 'PC', 5, 6);
INSERT INTO `categories` VALUES(12, 10, 8, 'MAC', 7, 8);
INSERT INTO `categories` VALUES(13, 8, 8, 'Printers', 10, 11);
INSERT INTO `categories` VALUES(14, 8, 8, 'Cameras', 12, 13);

How can I build an ordened list representing this tree?

With the sql bellow:

SELECT c. * , (COUNT( p.id ) -1) AS depth
FROM `categorias` AS p
CROSS JOIN categories AS c
WHERE (
c.lft
BETWEEN p.lft
AND p.rht
)
GROUP BY c.id
ORDER BY c.lft;

I got this result:

As you can see, I need to order by root_id too, so that I can generate the correct tree.

Also, after get the tree, is there a way to order each node by name?

I got it.

All you need to do is set root_id to the top parents too, so that you can ORDER BY correctly.

With the query bellow I can have separeted trees, and uptade only the tree that I'm working on:

SELECT c . * , count( p.id ) AS depth
FROM `categories` c
CROSS JOIN categories p
WHERE (
c.lft
BETWEEN p.lft
AND p.rht
)
AND c.root_id = p.root_id
GROUP BY c.id
ORDER BY c.root_id, c.lft