Chaining System Query in MySQL
Your Question
What you are looking to do is achievable in SQL Server using CTE recursive queries. MySQL's dialect of SQL does not support such querying.
I do understand the data and the output. For what you are looking to do, this is the solution:
R : 4
NodeQueue: &lt;Empty&gt;
OutputString: &lt;Empty&gt;

Add R's Children to NodeQueue
R : 4
NodeQueue: 2,10,11
OutputString: &lt;Empty&gt;

Remove Front of NodeQueue as R, Add R's Children to the NodeQueue, Add R to OutputString
R : 2
NodeQueue: 10,11,1
OutputString: 2

Remove Front of NodeQueue as R, Add R's Children to the NodeQueue, Add R to OutputString
R : 10
NodeQueue: 11,1,3
OutputString: 2,10

Remove Front of NodeQueue as R, Add R's Children to the NodeQueue, Add R to OutputString
R : 11
NodeQueue: 1,3,35
OutputString: 2,10,11

Remove Front of NodeQueue as R, Add R's Children to the NodeQueue, Add R to OutputString
R : 1
NodeQueue: 3,35,34
OutputString: 2,10,11,1

Remove Front of NodeQueue as R, Add R's Children to the NodeQueue, Add R to OutputString
R : 3
NodeQueue: 35,34 (3 has no children to add)
OutputString: 2,10,11,1,3

Remove Front of NodeQueue as R, Add R's Children to the NodeQueue, Add R to OutputString
R : 35
NodeQueue: 34 (35 has no children to add)
OutputString: 2,10,11,1,3,35

Remove Front of NodeQueue as R, Add R's Children to the NodeQueue, Add R to OutputString
R : 34
NodeQueue: &lt;Empty&gt; (35 has no children to add)
OutputString: 2,10,11,1,3,35,34

Algorithm ends here because the NodeQueue became empty.
GOOD NEWS
I have already written MySQL Stored Functions that do this

Oct 24, 2011 : Find highest level of a hierarchical field: with vs without CTEs
Dec 10, 2012 : MySQL: Tree-Hierarchical query

Please look over these over because the algorithm I described is in these two posts. You will have to decide what constitutes the Root to start with.
Give it a Try !!!