174 Mr. A. Cayley on the Theory of the 



third tree, and the fourth tree of fig. 3. To derive fig. 3 {bis) 

 from fig. 3, we must fill up the trees of fig. 3 with U at the root 

 and E,, Q, P at the other knots in every possible manner, subject 

 only to the restriction, that, reckoning up from the extremity of 

 a free branch to the root, there must not be any transposition 

 in the order of the symbols RQP, and taking care to admit only 

 distinct trees. Thus the first tree of fig. 3 might be filled up in 

 six ways ; but the trees so obtained are considered as one and 

 the same tree, and we have only the first tree of fig. 3 {bis). 

 Again, on account of the restriction, the fourth tree of fig. 3 can 

 be filled up in one way only, and we have thus the sixth tree of 

 fig. 3 {bis). And thus, in general, each figure of the second set 

 can be formed at once from the corresponding figure of the first 

 set j or when the first set of figures is given, the expression for 

 YX . . QPU can be formed directly without the assistance of the 

 expression for the preceding symbol X . . . QPU ; the number 

 of terms for the nth figure of the second set is obviously 

 1 . 2 . 3 ... n, and consequently it is only necessary to count the 

 terms in order to ascertain that no admissible mode of filling up 

 has been omitted. 



The number of parts in any one of the figures of the first set 

 is much smaller than the number of parts in the corresponding 

 figure of the second set ; and the law for the number of parts, 

 i. e. for the number A„ of the trees with 7i branches, is a very 

 singular one. To obtain this law, we must consider how the 

 trees with n branches can be formed by means of those of a 

 smaller number of branches. A tree with n branches has either 

 a single main branch, or else two main branches, three main 

 branches, &c. ... to n main branches. If the tree has one main 

 branch, it can only be formed by adding on to this main branch 

 a tree with n — l branches, i. e. A„ contains a part A„_i. If the 

 tree has two main branches, then ]3 + q being a partition of n — 2, 

 the tree can be formed by adding on to one main branch a tree 

 of J9 branches, and to the other main branch a tree of q branches ; 

 the number of trees so obtained is A^ A, : this, howevei-, assumes 

 that the parts ]} and q are unequal ; if they are equal, it is easy 

 to see that the number of trees is only |^Ap(Ap+l). Hence 

 p + q being any partition of n — 2, A„ contains the part ApA, if 

 p and q are unequal, and the part ^Ap{Ap + l) Up and q are 

 equal. In like manner, considering the trees with three main 

 branches, then ii p + q + 7- is any partition of ra — 3, A„ contains 

 the part A^A^A^ if p, q, r are unequal ; but if two of these 

 numbers, e. g. j) and q, are equal, then the part iAp(A^ + l)Ar; 

 and \i p, q, r are all equal, then the part ^Ap(Ap + l)(Ap + 2) ; 

 and so on, until lastly we have a single tree with n main branches, 

 or A„ contains the part unity. A little consideration will show 

 that the preceding rule for the formation of the number A» ia 



