ON THE ANALYTICAL FORMS CALLED TEEES. 261 



and the inimber of centre- or biccntre-trees is al\vays = l : viz. n odd, there ia 

 one centre-tree ; and n even, one biccntre-trce ; it is only considered as a par- 

 ticular case of the general theorem. The case where the number is =3 is ana- 

 lytically interesting : although there may not exist, for any 3-valeut element, 

 a series of hydrogen compounds B„ H„_,_2 corresponding to the paraffiues. The 

 case -svliero the number is =4, or say the carbon-trees, is that which pre- 

 sents the chief chemical interest, as giving the paraffines C„ Ho„^.2 ; and I caU 

 to mind here that the theory of the carbon-root trees is established as an 

 analytical result for its own sake and as the foundation for the other case, 

 but that it is the number of the carbon centre- and bicentre-trees which is 

 the number of the paraffines. 



The theory extends to the case where the number of branches from a knot 

 is at most = 5, or = any larger number ; but I have not developed the 

 formula. 



I pass now to the analytical theory : considering first the case of general 

 root-trees, we endeavour to find for a given altitude N the number of trees 

 of a given number of knots w and main branches a, or say the generating 

 function 



where the coefficient il gives the number of the trees in question. And we 

 assume that the problem is solved for the cases of the several inferior alti- 

 tudes 0, 1, 2, 3 . . . ]sr-i. 



This being so, observe that a tree of altitude N can be built up as shown 



ill the figure (which I call the edification diagram), by combining one 

 or more trees of altitude N" — 1 with a single tree of altitude not 

 exceeding I^ — 1 ; viz. in the figure, K = 3, we have the two trees 

 a, b, each of altitude 2, combined (as shown by the dotted lines) with 

 the tree c of altitude 1 : the whole number of knots in the resulting tree is 

 the sum of the number of knots on the three trees a, h, c : the number of 

 main branches is equal to the number of the trees a, h, plus tlie luimber of 



