428 
[610 
ON THE ANALYTICAL FORMS CALLED TREES, WITH 
completely determined by means of the tree formed with the carbon-atoms, or say of 
the carbon-tree, and that the question of the determination of the theoretic number 
of the paraffins is consequently that of the determination of the number of 
the carbon-trees of n knots, viz. the number of trees with n knots, subject to the 
condition that the number of branches from each knot is at most = 4. 
In the paper of 1857, which contains no application to chemical theory, the 
number of branches from a knot was unlimited; and, moreover, the trees were 
considered as issuing each from one knot taken as a root, so that, n = o, the trees 
regarded as distinct (instead of being as above only 3) were in all 9, viz. these were 
which, regarded as issuing from the bottom knots, are in fact distinct; while, taking 
them as issuing each from a properly selected knot, they resolve themselves into the 
above-mentioned 3 forms. The problem considered was in fact that of the “ general 
root-trees with n knots”—general, inasmuch as the number of branches from a knot 
was without limit; root-trees, inasmuch as the enumeration was made on the principle 
last referred to. It was found that for 
knots 1, 2, 3, 
No. of trees was... 1, 1, 2, 
= 1> A x , A 2 , 
the law being given by the equation 
(1 — X) _1 (1 — X 2 )~ A ' (1 — X 3 )~ A * (I — X*)- A * 
4, 5, 6, 
4. 9, 20, 
■A-3, A±, A 5 , 
7, 
8, 
48, 
115,. 
A 6 , 
A ? , 
= 1 + A l x + Ao x 2 + A 3 of + A a cd 4 + ... ; 
but the next following numbers A 8 , A 9 , A w , the correct values of which are 286, 
719, 1842, were given erroneously as 306, 775, 2009. I have since calculated two 
more terms, A u , A 12 = 4766, 12486. 
The other questions considered in the paper of 1857 and in that of 1859 have 
less immediate connexion with the present paper, but for completeness I reproduce 
the results in a Note*. 
* In the paper of 1857 I also considered the problem of finding B r the number with r free branches, 
with bifurcations at least: this was given by a like formula 
leading to 
for 
(1 - a;) -1 ( 1 - x 2 )~ B * (1 - x 3 )- 3 * (1 - x 4 )~ Bl ... = l + x+2B 2 x 2 +2B 3 x 3 + 2B 4 x 4 ..., 
B r = 1, 2, 5, 12, 33, 90 
f— 2, 3, 4, 5, 6, 7, 
In the paper of 1859, the 
knots : we have here 
question is to find the number of trees with a given number m of terminal 
(fm=l. 2.3...(m - 1) coefficient of x m ~ l in ^ ^ ,