88 BELL SYSTEM TECHNICAL JOURNAL 



Theorem 10: It is possible to construct m dijferent trees in the same n variables 

 with the folknving distribution: 



ai , a2 , ■ ■ ■ , an = (ni, 2m, 4m, ■■ ■ , 2''~'^m) 



It is interesting to note that under these conditions the bothersome 1 disap- 

 pears for m > 1. We can equalize the load on all n of the variables, not just 

 n — 1 of them, to within, at worst, one transfer element. 



7. The Function n{n) 



We are now in a position to study the behavior of the function m('0- 

 This will be done in conjunction with a treatment of the load distributions 

 possible for the general function of n variables. We have already shown 

 that any function of three variables can be realized with the distribution 



1,1,2 



in terms of transfer elements, and, consequently ix{i) < 4. 



Any function of four variables can be realized with the distribution 



1, 1, (2, 4) 

 Hence m(4) < 6. For five variables we can get the distribution 



1, 1, (2, 4, 8) 

 or alternatively 



1, 5, 5, (2, 4) 

 so that /i(5) < 10. With six variables we can get 



1, 5, 5, (2, 4, 8) and m(6) < 10 

 for seven, 



1, 5, 5, (2, 4, 8, 16) and m(7) < 16 



etc. Also, since we can distribute uniformly on all the variables in a tree 

 except one, it is possible to give a theorem analogous to Theorem 7 for the 

 function m('0- 



Theorem 11: For all n 



m(") < ^^ 



n+3 



For almost all n 



2' 

 m(") < - 



