¿Cómo selecciono solo una generación de un árbol jerárquico usando una relación padre-hijo de tabla con SQL?

Esto es detallado y feo y será lento, y está limitado a 4 generaciones, pero no sé de qué otra manera se puede hacer.

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 = ?