82 BELL SYSTEM TECHNICAL JOURNAL 



A tree with four bays can be constructed with any of the following dis- 

 tributions 



and the variables may be interchanged in any manner. The "sums" on the 

 right show how these distributions are obtained. The first set of numbers 

 represents the upper half of the tree and the second set the lower half. They 

 are all reduced to the sum of sets 1, 2, 4 or 1, 3, 3 in some order, and these 

 sets are obtainable for trees with 3 bays as we already noted. In general it is 

 clear that if we can obtain the distributions 



fli , flj ) fl3 > • • • > fln - . 



bi ibi f bi ^ ' • • , bn 



for a tree with n bays then we can obtain the distfibution 



1, fll + ^1 , ^2 + ^2 , • ' ' > a„ + bn 



for a tree with n + 1 bays. 



Now note that all the distributions shown have the following property: 

 any one may be obtained from the first, 1, 2, 4, 8, by moving one or more 

 units from a larger number to a smaller number, or by a succession of such 

 operations, without moving any units to the number 1. Thus 1, 3, 3, 8 is 

 obtained by moving a unit from 4 to 2. The set 1, 4, 5, 5 is obtained by 

 first moving two units from the 8 to the 2, then one unit to the 4. Further- 

 more, every set that may be obtained from the set 1, 2, 4, 8 by this process 

 appears as a possible distribution. This operation is somewhat analogous 

 to heat flow — heat can only flow from a hotter body to a cooler one just as 

 units can only be transferred from higher numbers to lower ones in the above. 



These considerations suggest that a disjunctive tree with u bays can be 

 constructed with any load distribution obtained by such a flow from the 

 initial distribution 



1,2,4,8, ••• ,2-> 

 We will now show that this is actually the case. 



