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THE BELL SYSTEM TECHNICAL JOURNAL, JULY 1951 



Proof. Every linear graph can be constructed by the following process. 

 Let the branches be numbered 1, 2, • • • , i^ and let the end points of the 

 k^^ branch be called Ak and Bk . There are iT — 1 places where partition 

 marks can be inserted in the sequence Ai, • • • , Ak and hence there are 

 2^"^ ways of partitioning the ^^'s into groups of the form 



Gi = Ui, ^2, ••• , Aa) 



G2 = {Aa+1 y '" yAb) 



Gz = (Ab+i y "' ,Ac) 





(a) 



(b) 



(c) 



i> 



(d) 



30 



(f) 



Fig. 7 — Examples of graphs. 



There are 2 ways of selecting some of the terminals 1,2 • • • , N. Suppose 

 that m of the terminals have been selected; then pick one of the partitions 

 of the AkS which has m or more groups Gi , • • • , Gm+« . Connect all the 

 end points in Gi to the first selected terminal, all the end points in G2 to 

 the second selected terminal, etc. Next connect the terminals in Gm+i , * • • , 

 Gm+f together to form 5 nodes. The number of ways of performing all these 

 operations is less than 



