SQL: get item level (aka generation) from parent child hierarchy

CREATE TABLE "public"."parent_child" (
    "id" int4 NOT NULL DEFAULT nextval('people_id_seq'::regclass),
    "first_name" varchar(100),
    "parent_id" int4,
    "city" varchar(100),
    PRIMARY KEY ("id")
);

id	first_name	parent_id	city
1	alex		washington
2	fred		washington
3	zoe	1	texas
4	mary	1	texas
5	brit	2	new mexico
6	joe	2	new mexico
7	bill	4	new mexico
8	anna	4	new mexico
9	jessy	8	new mexico
10	kriss	4	new mexico
11	stephen	5	new mexico
12	june	6	new mexico
13	suzy	6	miami
14	josephine	9	miami
15	sarah	9	miami
16	joni	9	miami

Get all generations

WITH RECURSIVE generation AS (
    SELECT
    	id,
    	first_name,
    	city,
    	parent_id,
    	0 AS generation_number
    FROM parent_child
    WHERE parent_id IS NULL
 
UNION ALL 
    SELECT
    	child.id,
    	child.first_name,
    	child.city,
    	child.parent_id,
    	generation_number +1 AS generation_number
    FROM parent_child child
    JOIN generation g ON g.id = child.parent_id
)

SELECT * FROM generation
ORDER BY generation;

Get some specific generation

WITH RECURSIVE generation AS (
    SELECT
    	id,
    	first_name,
    	city,
    	parent_id,
    	0 AS generation_number
    FROM parent_child
    WHERE parent_id IS NULL
 
UNION ALL 
    SELECT
    	child.id,
    	child.first_name,
    	child.city,
    	child.parent_id,
    	generation_number +1 AS generation_number
    FROM parent_child child
    JOIN generation g ON g.id = child.parent_id
)


SELECT * FROM generation
WHERE city = 'texas'
AND generation_number = 1
ORDER BY generation_number;