PostgreSQL: recursively join a second table - postgresql

Question

I'm struggling with recursion in PostgreSQL. I need to join a first table with a second one, and then recursively join within the second table. I looked at quite a number of examples, but most are about finding the parent records within a single table, and this has left me utterly confused.
Here's a minimal example with tables thing and category. Records in thing may or may not have a category:
id
name
category
1
a5
3
2
passat
2
3
apple
NULL
Records in category may have one or more parents in the same table:
id
name
parent_category
1
vehicle
NULL
2
car
1
3
coupe
2
The result I'm looking for is the combination of all things with their categories, as well as the category level (1 for the direct parent, 2 for the level above).
thing_name
category_name
level
a5
coupe
1
a5
car
2
a5
vehicle
3
passat
car
1
passat
vehicle
2
apple
NULL
NULL
I have a DB Fiddle here: https://www.db-fiddle.com/f/b7V8ddragZZ9x2RsMkdFYn/5
CREATE TABLE category (
id INT,
name TEXT,
parent_category INT
);
INSERT INTO category VALUES (1, 'vehicle', null);
INSERT INTO category VALUES (2, 'car', 1);
INSERT INTO category VALUES (3, 'coupe', 2);
CREATE TABLE thing (
id INT,
name TEXT,
category INT
);
INSERT INTO thing VALUES (1, 'a5', 3);
INSERT INTO thing VALUES (2, 'passat', 2);
INSERT INTO thing VALUES (3, 'apple', null);

Answer 1

Use a CTE to join the tables, giving you a tree-like view of combined thing_categories, which you can then use with a normal recursive CTE.
with recursive join_thing_category as (
select thing.id as thing_id,
thing.name as thing_name,
thing.category as thing_category,
category.id as category_id,
category.name as category_name,
category.parent_category as parent_category
from thing left join category on thing.category=category.id
),
recursive_part(n) as (
select thing_id, thing_name, thing_category, category_id, category_name, parent_category, (0*parent_category) + 1 as level from join_thing_category
union all
select 1, thing_name, thing_category, cat.id category_id, cat.name, cat.parent_category as parent, level+1 as level from recursive_part rp cross join category cat
where cat.id=rp.parent_category
)
select thing_name, category_name, level from recursive_part order by 1, 2, 3 limit 1024;
thing_name
category_name
level
a5
car
2
a5
coupe
1
a5
vehicle
3
apple
passat
car
1
passat
vehicle
2
View on DB Fiddle
The (0*parent_category) + 1 as level bit is so that things with no category get NULL as their level instead of 1.