Map ChildID to root ParentID

Transact-SQL
2

Maybe like this:

create table #yourtable(
   childid int
  ,parentid int
);

insert into #yourtable(childid,parentid)
 values (1,1)
       ,(2,1)
       ,(5,1)
       ,(4,5)
       ,(3,2)
       ,(6,6)
       ,(9,6)
       ,(7,9)
       ,(8,7)
;

alter table #yourtable
  add rootid int
;

update t1
   set t1.rootid=isnull(t2.parentid,t1.parentid)
  from #yourtable as t1
       left outer join #yourtable as t2
                    on t2.childid=t1.parentid
                   and t2.childid<>t2.parentid
;

while (@@rowcount>0)
begin
   update t1
      set t1.rootid=t2.parentid
     from #yourtable as t1
          inner join #yourtable as t2
                  on t2.childid=t1.rootid
                 and t2.childid<>t2.parentid
   ;
end;

select * from #yourtable;

drop table #yourtable;