246 ON THE THEORY OF THE ANALYTICAL FORMS CALLED TREES. [203 
those in the annexed figure, and the number is therefore two. It is not difficult to 
see that we have in this case (B r being the number of such trees with r free branches), 
(1 — x)~ x (1 — (1 — ¿r 3 ) - - 83 (1 — x x )~ Bi ... = 1 + x + 2B 2 x- + 2B x xr + 2B i x i + &c.; 
and a like process of development gives : 
B r = 
for r = 
1 
2 
5 
12 
33 
90 
2 
3 
4 
5 
6 
7 
I may mention, in conclusion, that I was led to the consideration of the fore 
going theory of trees by Professor Sylvester’s researches on the change of the inde 
pendent variables in the differential calculus. 
2, Stone Buildings, January 2, 1856.