ON THE ANALYTICAL FORMS CALLED TREES. 2G5 



applied to it only for the sake of seeing what the general method becomes in 

 such an extreme case. 



We thus form tlie Tobies, ^Yhich I proceed to explain. 



Table I. of general root-trees is in fact a Table of triple entrj', \-iz. it gives 

 for anj' given number of knots from 1 to 13 the number of root-trees cor- 

 responding to any given number of main branches and to any given altitude. 

 In each compartment, that is for any given number of knots, the totals of the 

 columns give the number of the trees for each given altitude, and the totals 

 of the lines give the number of the trees for each given number of main 

 branches : the corner grand totals of these totals respectively show for each 

 given number of knots the -whole number of root-trees : — 



& 



viz. knots.... 1 2 3 4 5 6 7 8 9 10 11 12 13 

 numbers are. . 1 1 2 4 9 20 48 115 286 719 1842 4766 12486 



as already mentioned, and which numbers were calculated by an independent 

 method. 



Table II. of general centre- and bicentre-trees consists of a centre part and 

 a bicentre part : the centre part is arranged precisel}^ in the same manner as 

 the root-table. As to the bicentre part, where it will bo observed there is no 

 division for number of main branches, the calculation of the several columns 

 is effected by the before-mentioned formula, 



thus column 2, we have by Table I. (totals of column 2) 



<^.v = x' + 2x* + 4x' 4- 6x' + 10.r^ + Ux' -\-21x'+29x'°+ . . . , 



and thence 



0,.^■=.^•'' + 2.v^ + Ix" + 14.x' + Q2x^'' + bSx'"' + 110a;'=-f 187.t-'' -{- . . 



'As already mentioned. Table I. is calculated each column by means of a 

 generating function given as a product of two factors, each of which is ob- 

 tained from the columns which precede the column in question ; and Table II., 

 the centre part of it, is calculated by means of the same generating functions 

 slightly modified : these generating functions serving for the calculation of 

 the two Tables are given in the table entitled " kSubsidiary Table for the cal- 

 culation of the GP's of Tables I. and II.," which immediately follows these 

 two Tables, and will bo further explained. 



