Stringing together columns with UPDATE SQL

Sadly, there is nothing quite as elegant for T-SQL or PL/SQL as MySQL's wonderful GROUP_CONCAT function. The drop-dead simple solution using GROUP_CONCAT is:

select BudgetID 
     , group_concat(BudgetPath separator '>')
              as BudgetPaths
  from Budgets 
group
    by BudgetID

For Oracle, have a look at: Displaying multiple records in one row.

For T-SQL, here's a method that will work. First, create a temporary table like this:

create table TempBudgetPaths
( BudgetID    integer      not null primary key 
, BudgetPaths varchar(max) null
)

Next, populate this table with keys:

insert 
  into TempBudgetPaths
     ( BudgetID )
select distinct 
       BudgetID 
  from Budgets

Finally, update the temporary table like this:

DECLARE @row INTEGER, @maxrows INTEGER
SELECT @row=1, @maxrows=count(*) FROM Budgets
WHILE @row <= @maxrows
  BEGIN
  UPDATE TempBudgetPaths
     SET BudgetPaths 
       = COALESCE(t1.BudgetPaths+'>','')
                + t2.BudgetPath
    FROM TempBudgetPaths t1
  INNER
    JOIN (
         SELECT BudgetID
              , BudgetPath 
              , ROW_NUMBER() OVER 
                  ( ORDER BY BudgetID
                           , BudgetPath ) 
                         as rownum
           FROM Budgets 
         ) AS t2
      ON t2.BudgetID = t1.BudgetID
   WHERE t2.rownum = @row
  SELECT @row = @row + 1
  END
SELECT * FROM TempBudgetPaths

Not nearly as pretty as GROUP_CONCAT, is it. Nevertheless, it works, and produces these results:

BudgetID     BudgetPaths
   1          Americas>HR>HRIT
   2          HR>Staffing
   3          Asia Pacific>Finance>HR>IT

Notice the sequence in which the BudgetPaths were appended. This is a result of the OVER (ORDER BY ...) clause. This is not the same as your sequence, which appears to be some kind of hierarchy. To achieve your sequence will require an additional data column, to indicate sequence of BudgetPath within BudgetID.