MySQL 8.0

By using db<>fiddle, you agree to license everything you submit by Creative Commons CC0.
CREATE TABLE categories (
  id int(11) NOT NULL,
  parentId int(11) NOT NULL,
  name varchar(1000) NOT NULL,
  path varchar(1000) NOT NULL
)
INSERT INTO `categories` (`id`, `parentId`, `name`, `path`) VALUES
(1, 0, 'Cat 1', 'Cat 1'),
(2, 1, 'Cat 2', 'Cat 1 > Cat 2'),
(3, 2, 'Cat 3', 'Cat 1 > Cat 2 > Cat 3'),
(4, 0, 'Cat A', 'Cat A'),
(5, 4, 'Cat B', 'Cat A > Cat B');

Records: 5  Duplicates: 0  Warnings: 0


/* Build hierarchy until node is reached */
WITH RECURSIVE cte (id, name, parentId, path, lvl) AS (
  SELECT
     id,
     name,
     parentId,
     path,
     1
  FROM categories
  WHERE id = 3

  UNION ALL

  SELECT
     cat.id,
     cat.name,
     cat.parentId,
     cat.path,
     cte.lvl + 1
  FROM cte
  INNER JOIN categories cat
    ON cte.parentId = cat.id
)

SELECT id, name, parentId, path
FROM cte
ORDER BY lvl DESC
;
     


id
name
parentId
path


1
Cat 1
0
Cat 1

2
Cat 2
1
Cat 1 > Cat 2

3
Cat 3
2
Cat 1 > Cat 2 > Cat 3


/* Build hierarchy starting from node */
WITH RECURSIVE cte (id, name, parentId, path, lvl) AS (
  SELECT
     id,
     name,
     parentId,
     path,
     1
  FROM categories
  WHERE id = 2

  UNION ALL

  SELECT
     cat.id,
     cat.name,
     cat.parentId,
     cat.path,
     cte.lvl + 1
  FROM cte
  INNER JOIN categories cat
    ON cte.id = cat.parentId
)

SELECT id, name, parentId, path
FROM cte
ORDER BY lvl ASC
;


id
name
parentId
path


2
Cat 2
1
Cat 1 > Cat 2

3
Cat 3
2
Cat 1 > Cat 2 > Cat 3


/* Find unmatched path */
WITH RECURSIVE cte (root, id, name, parentId, old_path, new_path, lvl) AS (
  SELECT
      id,
      id,
      name,
      parentId,
      path,
      name,
      1
  FROM categories
  WHERE parentId = 0

  UNION ALL

  SELECT
      cte.root,
      cat.id,
      cat.name,
      cat.parentId,
      cat.path,
      CONCAT(cte.name, ' > ', cat.name),
      cte.lvl + 1
  FROM cte
  INNER JOIN categories cat
    ON cte.id = cat.parentId
)
  
SELECT *
FROM cte
WHERE old_path != new_path
ORDER BY root, lvl
;


root
id
name
parentId
old_path
new_path
lvl


1
3
Cat 3
2
Cat 1 > Cat 2 > Cat 3
Cat 2 > Cat 3
3
id	name	parentId	path
1	Cat 1	0	Cat 1
2	Cat 2	1	Cat 1 > Cat 2
3	Cat 3	2	Cat 1 > Cat 2 > Cat 3