ON THE THEORY OF THE ANALYTICAL FORMS CALLED TREES. 
may be effected very easily as follows: [the table as originally printed contained at the 
end of it some errors of calculation which were corrected, B.A. Report for 1875, p. 258]. 
A 1 = 
A,= 
A 4 
A r = 
a r = 
a 7 = 
Ao = 
A a = 
A r 
for r - 
1 
(1) 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
l 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
1 
(2) 
2 
3 
3 
4 
4 
5 
5 
5 
2 
2 
4 
4 
6 
6 
8 
8 
3 
3 
6 
6 
9 
4 
4 
j 
1 
1 
2 
(4) 
5 
7 
11 
13 
17 
23 
27 
4 
4 
8 
16 
20 
28 
44 
10 
10 
20 
1 
1 
2 
4 
(9) 
11 
19 
29 
47 
61 
91 i 
9 
9 
18 
36 
81 
99 
45 
1 
1 
2 
4 
9 
(20) 
28 
47 
83 
142 
235 
20 
20 
40 
80 
180 
1 
1 
2 
4 
9 
20 
(48) 
67 
123 
222 
415 ; 
48 
48 
96 
192 
1 
1 
2 
4 
9 
20 
48 
(115) 
171 
318 
607 
115 
115 
230 
1 
1 
2 
4 
9 
20 
48 
115 
(286) 
433 
837 
286 
286 
1 
1 
2 
4 
9 
20 
48 
115 
286 
(719) 
1123 
719 
1 
2 
4 
9 
20 
48 
115 
286 
719 
1842 
1 
2 
3 
4 
5 
6 
7 
8 
9 
10 
(1 - iC 2 )- 1 
(1 — iC 3 ) -2 
(1 - OC 6 )“ 20 
(1 - as 8 ) -115 
(1 — iC 9 ) -306 
(1 - X 10 )- 775 
I have had occasion, for another purpose, to consider the question of finding the 
number of trees with a given number of free branches, bifurcations at least. Thus, 
when the number of free branches is three, the trees of the form in question are 
Fig.