895] 
A THEOREM ON TREES. 
27 
4—2 
to be a 3 f3y 8 (viz. the knot a occurs three times, and the knots /3, y, 8 each once). 
And so in other cases. 
This being so, I write 
(a + /3 4- 7 + 8 + e + £) 4 a/3y8e% = la 4 6 | 6 
+ 4 a s /3 30 120 
+ 6 a 2 /3 2 15 \ a(3y8e90 
+ 12 a 2 /3y 60 720 
+ 24 a(3y8 15 / 360 
1296, 
where the numbers of the left-hand column are the polynomial coefficients for the 
index 4; and the numbers of the right-hand column are the numbers of terms of 
the several types, 6 terms a 4 , 30 terms a 3 /3, 15 terms a 2 /3 2 , &c.: the products of the 
corresponding terms of the two columns give the outside column 6, 120, 90, &c.; 
and the sum of these numbers is of course 6 4 , = 1296. 
It is to be shown that we have 
1 tree a 4 . a/3y8e% (= a 5 /3ySe£); 4 trees a 3 /3 . a/3y8e% (= a 4 /3 2 y8e%),..., 
24 trees a(3y8 . a/3y8e% (= a 2 /3 2 7 2 S 2 e£): 
for this being so, then by the mere interchange of letters, the numbers 1, 4, 6, ... of 
the left-hand column have to be multiplied by the numbers 6, 30, 15, ... of the 
right-hand column., and we have the numbers in the outside column, the sum of 
which is = 1296 as above. 
Start with the last term a/3y8. a(3y8e%, = a 2 /3 2 y-8 2 e%. We have the trees 
eoi(3y8£ (= ea . a/3 . (3y . 7S . 8%), 
where the a, /3, 7, 8 may be written in any one of the 24 orders, and the number 
of such trees is thus = 24. If we consider only the 12 orders (a/3y8 being regarded 
as equivalent to 8y/3a), then the e, % may be interchanged; and the number is thus 
2 x 12, = 24 as before. 
Now for the 8 of a/3y8 substitute a, or consider the form afiya. a/3y8e£, = a 3 /3 2 y 2 8e£. 
We see at once in the form ea.. a/3. f3y. y8.8£, which one it is of the two S’s that 
must be changed into a: in fact, changing the first 8, we should have ea. a/3. /3y. yet. 8% 
which contains a circuit a(3y, and a detached branch 8£, and is thus not a 
tree: changing the second 8, we have ea . a/3 . /3y . y8. a£ which is a tree a s (3 2 y 2 8e£, 
= ag. ae . a/3 . f3y. y8. And similarly for any other order of the a/3y8, there is in each 
case only one of the 8’s which can be changed into a; and thus from each of the 
24 forms we obtain a tree 0?¡3 2 y 2 8e%. But dividing the 24 forms into the 12 + 12 
forms corresponding to the interchange of the letters e, £, then the first 12 forms, 
and the second 12 forms, give each of them the same trees a 3 j3 2 y 2 8e£; and the 
number of these trees is thus ^-.24, =12.