Mr. A. Cayley on the Analytical Forms called Trees. 375 



knots, three knots, and four knots respectively are shown in the 

 figures 1, 2, 3, and 4 : 



Fig. 4. 



•'ig. I. Fig. 2. 



Fig. 3. 



;vA 



and similarly for any number of knots. The trees with four 

 knots are formed first from those of one knot by attaching 

 thereto in every possible way (one way only) four knotted 

 branches j secondly, from those with two knots by attaching 

 thereto in every possible way (three different ways) four knotted 

 branches ; and thirdly, from those M'ith three knots by attaching 

 thereto in every possible way (three difi"erent ways) four knotted 

 branches, — the original knots of the trees of one knot and two 

 and three knots, being no longer terminal knots, are disregarded. 

 The total numbers of trees with one knot and with two and three 

 knots being respectively 1, 1, 3; the total number of trees with 

 four knots is 1.1 + 3.1 + 3.3=13. And in general, if the 

 number of trees with m knots is ^m, then it is easy to see that 

 we have 



(^OT = 01+ -^—(f)2+ —~ 03 



or what is the same thing, 



2<^m = 01 + — ^(^2+ .-- </)3...+ --</)(/;<— l) + 0?«. 



m — 1 . , ,. 



