36^ REPORT — 1875. 



main branches of the tree c. It is to be observed that the tree c may reduce 

 itself to the tree (.) of one knot and of altitude zero ; but each of the trees 

 a, h, as being of the altitude N— 1, must contain at least IST knots. 



Taking N=2 or any larger number, it is hence easy to see that the re- 

 quired generating function ^Q.f-x^ is 



= (1-^^)-^ (1-a'N+i)-'^ (l-teN+2)-'-' ...[«'•■•"] (first factor), 



X + {tyc" + {t, t-)x^ + (t,f, fi)x^ + . .. (second factor), 



where, as regards the first factor, the exponents taken -with reversed sign, that 

 is as positive, are l=no. of trees, altitude N — l,ofX knots; Zj=ditto, same alti- 

 tude, of (N+1) knots; ?.,=ditto, same altitude, of N + 2 knots, and so on ; and 

 where the symbol [i' ' ' ' °° ] denotes that in the function or product of factors 

 which precedes it, the tenns to be taken account of are those in f, t-,t^ . . . ; 

 viz. that the term in t°, or constant term ( = 1 in fact), is to be rejected. 



In the second factor the expressions .v, (t)x-, {t, f) x^, . . . represent (for given 

 exponents of t, x, denoting the number of main branches and the number of 

 knots respectively) the number of trees of altitude not exceeding N— 1 : thus 

 X, =lfx^ represents the number of such trees, 1 knot, main branch, 

 =1 ; and so if the value of (t, f, f, t^)x' be (at+ftt- + yf + H')x% then for 

 trees of an altitude not exceeding IS" — 1, and of o knots, a represents the 

 number of trees of 1 main branch, /3 that of trees of 2 main branches, y that 

 of trees of 3 main branches, d that of trees of 4 main branches. It is clear 

 that the number of trees satisfj'ing the given conditions and of an altitude not 

 exceeding N — 1 is at once obtained bj' addition of the numbers of the trees 

 satisfying the given conditions, and of the altitudes 0,1,2 ... X—1 ; aU 

 which numbers are taken to be known. 



It is to be remarked that the first factor, 



(l-<.r^)-' (l-f,t'N+')-'i {l-ti^+'~y-^ . . [«'••"], 



shows by its development the number of combinations of trees a, 6 . . of the 

 altitude X — 1 ; one such tree at least viust be taken, and the symbol [«'•••"=] 

 gives effect to this condition : the second factor x + (t)x' + (t, f) x^ + . . shows 

 the number of the trees c of altitude not exceeding X — 1. And this being 

 so, there is no difficiilty in seeing how the product of the two factors is the 

 generating function for the trees of altitude X. 



In the case ^=0, the generating function, or GF, is =x ; viz. altitude 0, 

 there is only the tree (.) 1 knot, main branch. 



N = 1, the GP is= (1 - te)~ '[<>■•"] a,-, = tx' + fx' + fx' , 



viz. altitude 1, there is 1 tree tx'^, 2 knots, 1 main branch ; 1 tree fx^, 3 knots, 

 2 main branches ; and so on. 

 Hence ^=2, we obtain 



GF = (1 - te=)- ' (1 - tx') - ' (1 - /.t'*) - ' ... [if '•••"]. (.V + a-- + fx^ + « V ....); 



viz. as regards the second factor, altitude not exceeding 1, that is =0 or 1, 

 there is altitude 0, 1 tree x, and altitude 1, 1 tree tx^, 1 tree fx^, and soon. 

 And we hence derive the GF's for the higher values N=3, 4, &c. : the de- 

 tails of the process will be afterwards more fully explained. 



