Recursively counting the number of objects in a directory with a materialized path, the best solution?

Question

PostgreSQL

There is a directory of the form (id, path, name) in the directory products (id, categoryId, name)

categories
====================

|id| path      |name
|1 | {1}       | Root1
|2 | {2}       | Root2
|3 | {1,3}     | Cat1_3
|4 | {1,4}     | Cat1_4
|5 | {1,3,5}   | Cat1_3_5
|6 | {1,4,6}   | Cat1_4_6
|7 | {1,3,5,7} | Cat1_3_5_7

products
====================

id| categoryId     |name
1 | 3              | p3_1
2 | 3              | p3_2
3 | 6              | p6_1
4 | 6              | p6_2
5 | 7              | p7_1
6 | 7              | p7_2
7 | 7              | p7_3
8 | 2              | p2_1

The task is to display subcategories for a given one with the number of products including all subcategories - expected result
[Root] -
id| path |name | product_count
1 | {1} | Root1, | 7
2 | {2} | Root2, | 1
[1]
id| path |name | product_count
3 | {1,3} | Cat1_3 | 5
4 | {1,4} | Cat1_4 | 2
[1,3]
id| path |name | product_count
5 | {1,3,5} | Cat1_3_5 | 3
[1,4]
id| path |name | product_count
6 | {1,4,6} | Cat1_4_6 | 2
In principle, there is a working solution,

WITH product_counter AS
(SELECT
   "loc_categories"."path",
   coalesce(COUNT("loc_products"."id"), 0)  as cnt
 FROM "loc_products"
   LEFT JOIN loc_categories ON "loc_products"."categoryId" = "loc_categories"."id"
 WHERE "loc_products"."amount" > 0
 GROUP BY  "loc_categories"."path"
)
SELECT
  "loc_categories"."id",
  "loc_categories"."name",
   SUM("pcnt"."cnt") as product_count
FROM "loc_categories"
  LEFT JOIN
  product_counter AS pcnt ON (ARRAY [loc_categories."id"] && pcnt.path)

WHERE ("loc_categories"."path" @> :inputPath AND array_length(loc_categories.path,1) =:inputLevel)
GROUP BY loc_categories."id"
HAVING SUM("pcnt"."cnt") > 0

And the question is, is it possible to solve the problem somehow more beautifully?