Hur väljer jag bara en generation av ett hierarkiskt träd med en underordnad tabellrelation med SQL?

Det här är mångsidigt och fult och kommer att gå långsamt, och det är begränsat till fyra generationer, men jag vet inte hur det annars kan göras.

SELECT person_name, parent_name FROM

    (SELECT child1.name AS person_name, parent1.name AS parent_name, '1' AS generation
    FROM people as child1
    JOIN people as parent1
    ON child1.parent_person_id = parent1.person_id AND child1.parent_person_id = child1.person_id

    UNION

    SELECT child2.name AS person_name, parent2.name AS parent_name, '2' AS generation
    FROM people as child2
    JOIN people as parent2
    ON child2.parent_person_id = parent2.person_id AND child2.parent_person_id <> child2.person_id AND parent2.parent_person_id = parent2.person_id

    UNION

    SELECT child3.name AS person_name, parent3.name AS parent_name, '3' AS generation
    FROM people as child3
    JOIN people as parent3
    ON child3.parent_person_id = parent3.person_id AND parent3.parent_person_id <> parent3.person_id
    JOIN people as grandparent1
    ON parent3.parent_person_id = grandparent1.person_id AND grandparent1.parent_person_id = grandparent1.person_id

    UNION

    SELECT child4.name AS person_name, parent4.name AS parent_name, '4' AS generation
    FROM people as child4
    JOIN people as parent4
    ON child4.parent_person_id = parent4.person_id AND parent4.parent_person_id <> parent4.person_id
    JOIN people as grandparent2
    ON parent4.parent_person_id = grandparent2.person_id AND grandparent2.parent_person_id <> grandparent2.person_id
    JOIN people as greatgrandparent
    ON grandparent2.parent_person_id = greatgrandparent.person_id AND greatgrandparent.parent_person_id = greatgrandparent.person_id
    ) AS tree

WHERE generation = ?