Missing "public" schema when converting a "regclass" value to text
This is not related to the schema public per se - which is just another schema with no special powers except that it's created by default - and included in the search_path by default.

The current search_path is behind this. The text representation of a regclass value builds on it. The table name is only schema-qualified if the schema is not the first match in the search_path anyway. The manual:


  The regclass input converter handles the table lookup according to the
  schema path setting, and so it does the “right thing” automatically.
  Similarly, casting a table's OID to regclass is handy for symbolic
  display of a numeric OID.


You could force Postgres to print the schema name by setting an empty search_path (locally):

SET LOCAL search_path = '';
SELECT 'public.tt'::regclass::text;


But that forces you to schema-qualify everything for the rest of the transaction or until you set the search_path again. Well, most everything, pg_catalog and pg_temp are special. See:


How does the search_path influence identifier resolution and the “current schema”


Or you avoid the special cast and the search_path and get the schema name from pg_class directly:

SELECT relnamespace::regnamespace::text
FROM   pg_catalog.pg_class
WHERE  oid = 'public.tt'::regclass;