676 THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



The Tree 

 The number of contacts used in the tree of Fig. 3 is 



A ••• ^.(l + i+ — L- + ••• + -— L—). 



\ pk pk-1 pk pk " • pll 



It can be shown that the most economical way to build the tree is to put 

 the small switches at the narrow end of the tree. If the smallest number of 

 positions of any of the switches in the tree is p\ then the number of contacts 

 in the tree is less than 



Upper Bound 



Having counted the number of contacts which are used in the tree and 

 in the network which produces all functions in Fig. 4, it only remains to 

 decide how many of the given switches X\^ • • • , ^Cm are to be put in each 

 of these two parts. 



Theorem II. Let P be the largest of the numbers pi ^ • • • , pM of values which 

 the variables Xi , • • • j Xm can assume. Then any switching function of 

 (xi , • • • , Xm) can be synthesized using no more than 



contacts when H > 4 bits. 



To prove the theorem we consider two cases according as P is greater or 

 less than F - 2 log H. 

 Case 1: (P >' H - 2\ogH) 



In this case we use the synthesis process described above, putting all the 

 switches into the tree and none in the network which produces all functions. 

 The number of contacts used is less than 2-2^ and the theorem follows 

 because 



P>H-2\ogH, 



Case 2: {P < H - 2\ogH) 



In this case we use the synthesis process described above, putting into 

 the right-hand network a collection 5 of switches so chosen that YLs pi comes 

 as close as possible to -^ — 2 log ^ without actually exceeding it. Then if 



11^= {H - 2 log H)F, 



