ON THE ANALYTICAL FORMS CALLED TREES. 



259 



To count the trees on tlie principle first referred to, we require the notions 

 of " centre " and " biceutre," due, I believe, to Sj'lvcster ; and to establish 

 these we require the notions of "main branch" and "altitude": viz. in a 

 tree, selecting- any knot at pleasure as a root, the branches which issue from 

 the root, each with all the branches that belong to it, are the main branches, 

 and the distance of the furthest knot, measured by the number of iuterme- 



W V 



diate branches, is the altitude of the main branch. Thus in the left-hand 

 figure, taking A as the root, there are 3 main branches of the altitudes 3, 3, 1 

 respectively : in the right-hand figure, taking A as the root, there are 4 

 main branches of the altitudes 2, 2, 1, 3 respectively ; and we have then the 

 theorem that in every tree there is either one and only one centre, or else 

 one and only one bicentre ; viz. we have (as in the left-hand figure) a centre 

 A which is such that there issue from it two or more main branches of alti- 



(l-a;)-i (1 -a;=)-B2 (1 -.a3)-^3 (1 _^)-B3 . , . =1 +^+2B2 .rS+SB, .r3+2B4 a?4 . . ., 

 leading to 



Br= 1, 



forr= 2, 



2, 6, 12, 33, 90 



3, 4, 5, 6, 7 



In the paper of 1860, the question is to find the number of trees with a given number 

 m of terminal knots : we have here 



ipm=l .2.3... (w-1) coefficient x'^-^ in ^^-^> 



giving the values 



^m= 1, 1, 3, 13, 75, 541, 4683, 47293, 



for m= 1, 2, 3, 4, 5, 6, 7, 8, 



Eut if from each uon-terminal knot there asceucl two and only two branches, then in this 



_, . 1— 'V^l— 4a' 

 case 0>w= coefficient it'" 'in — ■^' 



pm=- 



2x 

 1.3.5..2M-3 



viz. we have the very simple form 

 2m- 1, 



1 . 2 . 3 . . . »re 



* giving ^m— 1, 1, 2, 5, 14, 42 



for m= 1, 2, 3, 4, 5, 7 



S2 



