CTE: Get all parents and all children in one statement
If you want all ancestors and all descendants, you can combine the two queries in one. Use the two CTEs and then a simple UNION:

WITH RECURSIVE 
    -- descendants 
    rec_d (id, name) AS
    (
      SELECT tree.id, tree.name FROM tree WHERE name = 'Father'
      UNION ALL
      SELECT tree.id, tree.name FROM rec_d, tree where tree.parent_id = rec_d.id
    ),
    --  ancestors
    rec_a (id, name, parent_id) AS
    (
      SELECT tree.id, tree.name, tree.parent_id FROM tree WHERE name = 'Father'
      UNION ALL
      SELECT tree.id, tree.name, tree.parent_id FROM rec_a, tree WHERE tree.id = rec_a.parent_id
    )
SELECT id, name FROM rec_a
UNION 
SELECT id, name FROM rec_d ;


If we haven't got any errors above, we can them improve it:


change the final UNION to UNION ALL by putting the starting node(s) in only one of the CTEs.
use JOIN .. ON instead of the implicit joins.
fix mismatches between SELECT and CTE column lists.


The query becomes:

WITH RECURSIVE 
    -- starting node(s)
    starting (id, name, parent_id) AS
    (
      SELECT t.id, t.name, t.parent_id
      FROM tree AS t
      WHERE t.name = 'Father'          -- this can be arbitrary
    ),
    descendants (id, name, parent_id) AS
    (
      SELECT s.id, s.name, s.parent_id 
      FROM starting AS s
      UNION ALL
      SELECT t.id, t.name, t.parent_id 
      FROM tree AS t JOIN descendants AS d ON t.parent_id = d.id
    ),
    ancestors (id, name, parent_id) AS
    (
      SELECT t.id, t.name, t.parent_id 
      FROM tree AS t 
      WHERE t.id IN (SELECT parent_id FROM starting)
      UNION ALL
      SELECT t.id, t.name, t.parent_id 
      FROM tree AS t JOIN ancestors AS a ON t.id = a.parent_id
    )
TABLE ancestors
UNION ALL
TABLE descendants ;