Let x be the last of these t' children of the 2nd child of the root. Let C,k,n,m be as given. The following is of independent interest. By induction on k. Hence C is of the required form.
C starts off by visiting the root.
Hence C is optimal. We will be content to verify only the second and third claims by induction. A standard presentation of the Ackerman hierarchy is as follows. We have made no attempt to optimize the expression 5n.