Analytical Forms called Trees. 175 



completely expressed by the equation 



= 1 + A,a^ + A2.^'2 ^ ^^^ ^ ^43^4 ^ &(._^ 



and consequently that we may, by means of this equation, calcu- 

 late successively for the different values of n the number A„ of 

 the trees with n branches. The calculation may be effected very 

 easily as follows : — 



A,= 



A = 



A,= 



A5 = 



Ae= 



A 



A 



A 



7 — 



A,: 



for r- 



1 (1) 1 1 1 1 1 1 



1111 1 1 



1111 



1 1 



11 (2) 233 4 4 5 5 



2 2 4 4 6 6 8 



3 3 6 6 



4 



1 1 2 (4) 5 7 11 13 17 23 27 



4 4 8 16 20 28 44 



10 10 20 



1 1 2 4 (9) 11 19 29 47 6] 91 



9 9 18 36 81 99 



36 36 



112 4 9 (20) 28 47 103 178 226 

 20 20 40 80 180 



1 1 2 4 9 20 (48) 67 143 258 406 

 48 48 96 192 



1 1 2 4 9 20 48(115)191354 598 



115 115 330 



1 1 2 4 9 20 48 115(306)469 928 



306 306 



1 1 2 4 9 20 48 115 306(775)1234 



775 



1 2 



1 2 



4 9 20 



3 4 



48 115 306 775 



8 9 



(2009) 



10 



{\-x^) 



,2\-l 



(1-^^)-' 



.4^-* 



\-x' _ 



(1-^5) -9 



1-^6) -20 

 (l-^7)-''« 



(l-a;8)-"' 



(l-a;9)-'"' 

 (l_a?iO)-775 



