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

 Whence if 



which gives for (f)m the expression 



<f)m = l .2.3... (m-1) coeff. x'"-' in ^— ^ ; 



and the value of (pm might easily be obtained in an explicit form 

 in terms of the differences of the powers of zero. The values of 

 <f>m. are, for 



m = l, 2, 3, 4, 5, 6, 7, 8, &c. 



<j,m = l, 1, 3, 13, 75, 541, 4683, 47293. 

 In the foregoing problem, the number of branches descending 

 from a non-terminal knot is one, two, or more. But assume 

 that the number of branches descending from a non-terminal 

 knot is always two ; so that attending, as before, only to the 

 terminal knots, the trees with two knots, three knots, four knots 

 respectively are shown in the figures, 5, 6, and 7. 

 Fig. 5. Fig. 6. Fig. 7. 



This corresponds to the following problem in the theory of 



