26 
[895 
895] 
895. 
to be a 3 
And so i 
This 
A THEOREM ON TREES. 
[From the Quarterly Journal of Pure and Applied Mathematics, vol. xxm. (1889), 
pp. 376—378.] 
The number of trees which can be formed with n + 1 given knots a, /3, y, ... is 
= (ii+l) n_1 ; for instance n = ‘3, the number of trees with the 4 given knots a, ¡3, y, 
8 is 4 2 = 16, for in the first form shown in the figure the a, ¡3, y, 8 may be arranged 
12 + 4 = 16 
in 12 different orders (a/3y8 being regarded as equivalent to Sy/3a), and in the second 
form any one of the 4 knots a, /3, y, 8 may be in the place occupied by the a: 
the whole number is thus 12 + 4, =16. 
Considering for greater clearness a larger value of n, say n = o, I state the 
particular case of the theorem as follows: 
No. of trees (a, /3, y, 8, e, £) = No. of terms of (a + ¡3 + y + 8 + e + £) 4 a/3y8et„ = 6 4 , = 129 6, 
and it will be at once seen that the proof given for this particular case is applicable 
for any value whatever of n. 
I use for any tree whatever the following notation: for instance, in the first of 
the forms shown in the figure, the branches are a/3, /3y, y8; and the tree is said 
to be a/3 2 y 2 8 (viz. the knots a, 8 occur each once, but /3, y each twice); similarly 
in the second of the same forms, the branches are a/3, ay, a8, and the tree is said 
where th 
index 4; 
the several 
corresponding 
and the 
It is 
1 tree a 4 . 
for this 
the left-ha 
right-hand 
which is = 
Start 
where the 
of such tree 
as equivalen 
2 x 12, = 24 
Now 
We see at 
must be cha 
which cont 
tree: chang 
= a£\ ae . a/3 
case only or 
24 forms 
forms corres 
and the s 
number of