Get products of category plus all child categories till leaf nodes

We have a hierarchy which looks this:

Now the problem is that most products are only connected with 1 category, which is most of the time the lowest level.

I am trying to get ALL products below a given category. So If select one category, I want ALL products from current category and ALL categories below. So this didn't work:

SELECT "category"."id"
    ,"products"."name"
FROM "products"
INNER JOIN "product_categories" ON "products"."id" = "product_categories"."product_id"
INNER JOIN "categories" ON "product_categories"."category_id" = "categories"."id"
WHERE "categories"."parent_id" = 0

Anyone who can help me out? BTW, we are using PostgreSQL

EDIT:

We are using "PostgreSQL 9.3.5" And these are the tables for this case, you can see in my above query (and below ERD) which connections I would like to accomplish.

EDIT 2: This was my query for mysql (here we use taxonomies and not categories):

SELECT T2.id, T2.taxonomy, T2.parent, T2.terms_id, terms.term
FROM (
    SELECT
        @r AS _id,
        (SELECT @r := parent FROM term_taxonomies WHERE id = _id) AS parent,
        @l := @l + 1 AS lvl
    FROM
        (SELECT @r := 1, @l := 0) vars,
        term_taxonomies m
    WHERE @r <> 0) T1
JOIN term_taxonomies T2
ON T1._id = T2.id
JOIN terms ON T2.terms_id = terms.id
ORDER BY T1.lvl DESC;

But this only works for mysql