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recursion - Recursive mysql select?

I saw this answer and i hope he is incorrect, just like someone was incorrect telling primary keys are on a column and I can't set it on multiple columns.

Here is my table

create table Users(id INT primary key AUTO_INCREMENT,
    parent INT,
    name TEXT NOT NULL,
    FOREIGN KEY(parent)
    REFERENCES Users(id)
);


+----+--------+---------+
| id | parent | name    |
+----+--------+---------+
|  1 |   NULL | root    |
|  2 |      1 | one     |
|  3 |      1 | 1down   |
|  4 |      2 | one_a   |
|  5 |      4 | one_a_b |
+----+--------+---------+

I'd like to select user id 2 and recurse so I get all its direct and indirect child (so id 4 and 5).

How do I write it in such a way this will work? I seen recursion in postgresql and sqlserver.

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CREATE DEFINER = 'root'@'localhost'
PROCEDURE test.GetHierarchyUsers(IN StartKey INT)
BEGIN
  -- prepare a hierarchy level variable 
  SET @hierlevel := 00000;

  -- prepare a variable for total rows so we know when no more rows found
  SET @lastRowCount := 0;

  -- pre-drop temp table
  DROP TABLE IF EXISTS MyHierarchy;

  -- now, create it as the first level you want... 
  -- ie: a specific top level of all "no parent" entries
  -- or parameterize the function and ask for a specific "ID".
  -- add extra column as flag for next set of ID's to load into this.
  CREATE TABLE MyHierarchy AS
  SELECT U.ID
       , U.Parent
       , U.`name`
       , 00 AS IDHierLevel
       , 00 AS AlreadyProcessed
  FROM
    Users U
  WHERE
    U.ID = StartKey;

  -- how many rows are we starting with at this tier level
  -- START the cycle, only IF we found rows...
  SET @lastRowCount := FOUND_ROWS();

  -- we need to have a "key" for updates to be applied against, 
  -- otherwise our UPDATE statement will nag about an unsafe update command
  CREATE INDEX MyHier_Idx1 ON MyHierarchy (IDHierLevel);


  -- NOW, keep cycling through until we get no more records
  WHILE @lastRowCount > 0
  DO

    UPDATE MyHierarchy
    SET
      AlreadyProcessed = 1
    WHERE
      IDHierLevel = @hierLevel;

    -- NOW, load in all entries found from full-set NOT already processed
    INSERT INTO MyHierarchy
    SELECT DISTINCT U.ID
                  , U.Parent
                  , U.`name`
                  , @hierLevel + 1 AS IDHierLevel
                  , 0 AS AlreadyProcessed
    FROM
      MyHierarchy mh
    JOIN Users U
    ON mh.Parent = U.ID
    WHERE
      mh.IDHierLevel = @hierLevel;

    -- preserve latest count of records accounted for from above query
    -- now, how many acrual rows DID we insert from the select query
    SET @lastRowCount := ROW_COUNT();


    -- only mark the LOWER level we just joined against as processed,
    -- and NOT the new records we just inserted
    UPDATE MyHierarchy
    SET
      AlreadyProcessed = 1
    WHERE
      IDHierLevel = @hierLevel;

    -- now, update the hierarchy level
    SET @hierLevel := @hierLevel + 1;

  END WHILE;


  -- return the final set now
  SELECT *
  FROM
    MyHierarchy;

-- and we can clean-up after the query of data has been selected / returned.
--    drop table if exists MyHierarchy;


END

It might appear cumbersome, but to use this, do

call GetHierarchyUsers( 5 );

(or whatever key ID you want to find UP the hierarchical tree for).

The premise is to start with the one KEY you are working with. Then, use that as a basis to join to the users table AGAIN, but based on the first entry's PARENT ID. Once found, update the temp table as to not try and join for that key again on the next cycle. Then keep going until no more "parent" ID keys can be found.

This will return the entire hierarchy of records up to the parent no matter how deep the nesting. However, if you only want the FINAL parent, you can use the @hierlevel variable to return only the latest one in the file added, or ORDER BY and LIMIT 1


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