ON THE ANALYTICAL FORMS CALLED TREES. 263 



So far we have considered root-trees ; but rcf'erriug to the last diagram, it 

 is at once seen that the assumed root ■will be a centre, provided only that 

 (instead of, it may be, only a single tree a of the altitude N — 1), we take 

 always two or more trees of the altitude jS"— 1 to form the new tree of the 

 altitude N. Aud we give effect to this condition by simply writing in place 

 of [*'•••"] the new symbol [f^"""], which denotes that only the terms 

 f, f, <* . . . are to be taken account of; viz. that the terms in <" and f are to 

 be rejected. The component trees of the altitude N — 1 are, it is to be ob- 

 served, as before, root-trees ; hence the second factor of the generating func- 

 tion is unaltered : the theorem is that for the centre-trees of altitude N^ we 

 have the same generating function as for the root-trees, writing only [i^--"] 

 in place of [i' ■•"']. Or, what is the same thing, supposing that the first 

 fector, unaffected by either symbol, is 



= l+w''(at + fie+ ..) + .v''+' (at + (3'f+ ...) + .., 

 then affecting it with [«'■•'"] the value for the root- trees is 



=a.-N (at+l3('+ . . )^-.^^N+' {a't + (i'e+ ...)+.., 

 and affecting it with [f--- "] the value for the centre-trees is 



It thus appears how the fundamental problem is that of the root-trees, its 

 solution giving at once that of the ceutre-trees ; whereas we cannot conversely 

 solve the problem of the root-trees by means of that of the centre-trees. 



As regards the bicentre-trees, it is to be remarked that starting from a 

 centre-tree of altitude N-|-l with two main branches, then by simply striking 

 out the centre, so as to convert into a single branch the two branches which 

 issue from it, we obtain a bicentre-tree of altitude N. Observe that the alti- 

 tude of a bicentre-tree is measui'ed by that of the longest main branch from 

 A or B, not reckoning AB or BA as a main branch. Hence the number of 

 bicentre-trees, altitude ^N, is = number of centre-trees of two main branches, 

 altitude N + l. 



This is in fact the convenient formula, provided only the number of centre- 

 trees of two main branches has been calculated up to the altitude N -|- 1 ; but 

 we can find independently the number of bicentre-trees of a given altitude N : 

 the bicentre-tree is in fact formed by taking the two connected points A, B 

 each as the root of a root-tree altitude N (the number of knots of the bicentre- 

 tree being thus, it is clear, equal to the sum of the numbers of knots of the 

 two root-trees respectively) ; and it is thus an easy problem of combinations 

 to find the number of bicentre-trees of a given altitude JST. Write 



as the generating function of the root-trees of altitude N ; viz. for such trees, 

 l=no. of trees with N + 1 knots, /3=no. with N-f-2 knots, and so on; 

 then the generating function of the bicentre-trees of the same altitude N is 



