28 
A THEOREM ON TREES. 
896] 
[895 
And in like manner reducing the a/3y8 to a 2 ft 2 , a 3 /3 or a 4 , we obtain in each 
case the number of trees equal to the proper sub-multiple of 24, viz. 6, 4, 1 in the 
three cases respectively (for the last case this is obvious, viz. there is 1 tree 
a 6 /3y8e%, = aft . ay . a8 . ae . a£); and the subsidiary theorem is thus proved. Hence the 
original theorem is true: as already remarked, it- is easy to see that the proof is 
perfectly general. 
The theorem is one of a set as follows: 
Let (X, a, /3, y, ...) denote as above the trees with the given knots X, a, /3, y, ...; 
(A, + /x, a, /3, y, ...) the pairs of trees with the given knots X, /x, a, /3, y, ..., the 
knots X, /x belonging always to different trees; (X + /x + v, a, /3, 7,...) the triads of 
trees with the given knots X, /x, v, a, /3, 7, ..., the knots X, /i, v always belonging 
to different trees; and so on: then if i +1 be the number of the knots X, /x, v, ..., 
and n the number of the knots a, /3, 7, ..., the number of trees or pairs, or triads, 
&c., of trees is = (i + 1) (i + n + l) n_1 . In particular, if i = 0, then n being the number 
of knots a, /3, 7, ..., and therefore n + 1 the whole number of knots X, a, /3, 7, ..., 
the number of trees is = (w + l) n_1 as before. 
As a simple example, consider the pairs (X + /x, a, /3): here i = 1, n = 2, and we 
have (¿ + 1)(H» + l) n_1 = 2.4, = 8: in fact, the pairs of trees are 
(Xa, aft, /x), (X/3, fta, /x), (Xa, Xft, /x), 
(fxa, aft, X), (/x/3, fta, X), (/xa, /xft, X); (Xa, /xft), (Xft, /xa). 
We may arrange the trees (a, ft, 7, 8, e) as follows: 
(a, ft, 7, 8, e) = aft (ft, 7, 8, e); 125 = 4 x 1.4 2 = 64 
+ aft . ay (ft + 7, 8, e) +6x2.4' 48 
+ aft. ay. a8 (ft + y + 8, e) +4x3.4° 12 
+ aft. ay. a8 . ae +1 1 
125, 
A 
[From the 
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viz. to obtain the trees (a, ft, 7, 8, e), we join on the branch aft to any tree 
(ft, y, S, e): the branches aft, ay to any pair of trees (ft + 7, 8, e); the branches 
aft, ay, a8 to any triad of trees (ft + y + 8, e); and take lastly the tree aft. ay. a8. ae: 
the knots ft, 7, 8, e being then interchanged in every possible manner. The whole 
number of trees 125 is thus obtained as = 64 + 48 + 12 + 1; the theorem is of course 
and as usua 
have identic; 
perfectly general. 
or say 
The foregoing theory in effect presents itself in a paper by Borchardt, “ Ueber 
eine der Interpolation entsprechende Darstellung der Eliminations-Resultante,” Crelle, 
t. LVil. (1860), pp. Ill—121, viz. Borchardt there considers a certain determinant, 
composed of the elements 10, 12, ... , 1 n, 20, 21, 23, ..., 2n, nO, nl, ..., nn — 1, and 
represented by means of the trees (0, 1, 2, ..., n)\ the branches of the tree being 
the aforesaid elements, and the tree being regarded as equal to the product of the 
several branches: the number of terms of the determinant is thus =(?i+l) n_1 as 
Hence ] 
and 
above.