86 BELL SYSTEM TECHNICAL JOURNAL 



i 



2_] Ml = 2' — C] — 02 ^ 2' — fli — fl2 — 2 



and 



since 



<+3 

 3 1 



22 ^t = 2_/ <^i — Ol — «2 > 2'"^^ — 01 — 02 



ai , 02 , • • • , On = (2, 4, 8, • • • , 2") 



This proves the lemma . 



5. The Disjunctive Tree 



It is now easy to prove the following: 



Theorem 8: A disjunctive tree of n bays can be constructed with any dis- 

 tribution 



ai , 02 , • • • , On = (1, 2, 4, • • • , 2"-^). 



We may prove this jby induction. We have seen it to be true for n = 

 2, 3, 4. Assuming it for n, it must be true for » + 1 since the Lemma shows 

 that any 



fli , ^2 , • • • , fln = (2, 4, 8, • • • , 2") 



can be decomposed into a sum which, by assumption, can be realized for the 

 two branches of the tree. 



It is clear that among the possible distributions 



(1, 2, 4, • ■ • , 2"-0 



for the tree, an "almost uniform" one can be found for all the variables but 

 one. That is, we can distribute the load on (n — 1) of them uniformly 

 except at worst for one element. We get, in fact, for 



w = 1 1 



n= 2 1, 2 



n = 3 1, 3, 3 



n = 4 1, 4, 5, 5, 



n = 5 1, 7, 7, 8, 8, 



n = 6 1, 12, 12, 12, 13, 13 



n = 7 1, 21, 21, 21, 21, 21, 21 



etc. 



as nearly uniform distributions. 



6. Other Distribution Problems 



Now let us consider the problem of load distribution in series-parallel 

 circuits. We shall prove the following: 



