188 UNICURSAL PROBLEMS [CH. IX 



Geometrical Trees. Euler's original investigations were 

 confined to a closed network. In the problem of the maze it 

 was assumed that there might be any number of blind alleys 

 in it, the ends of which formed free nodes. We may now 

 progress one step further, and suppose that the network or 

 closed part of the figure diminishes to a point. This last 

 arrangement is known as a tree. The number of unicursal 

 descriptions necessary to completely describe a tree is called 

 the base of the ramification. 



We can illustrate the possible form of these trees by rods, 

 having a hook at each end. Starting with one such rod, we 

 can attach at either end one or more similar rods. Again, 

 on any free hook we can attach one or more similar rods, 

 and so on. Every free hook, and also every point where two 

 or more rods meet, are what hitherto we have called nodes. 

 The rods are what hitherto we have termed branches or paths. 



The theory of trees — which already plays a somewhat 

 important part in certain branches of modern analysis, and 

 possibly may contain the key to certain chemical and biological 

 theories — originated in a memoir by Cayley*, written in 

 1856. The discussion of the theory has been analytical rather 

 than geometrical. I content myself with noting the following 

 results. 



The number of trees with n given nodes is n n ~\ If A n is 

 the number of trees with n branches, and B n the number of 

 trees with n free branches which are bifurcations at least, 

 then 



(l-aO-^l-a 2 )"^ 1 -* 3 ) - ^ = l+A 1 oc + A ! ^ + A 3 !xi 3 + ..., 



(l-x)~ 1 Q.-a! i )- B '(l-a?)- B * = 1 +x + 2B.j^ + 2£ 3 « 3 + .... 



* Philosophical Magazine, March, 1857, series 4, vol. :m, pp. 172 — 176 ; or 

 Collected Works, Cambridge, 1890, vol. in, no. 203, pp. 242 — 216 : see also the 

 paper on double partitions, Philosophical Magazine, November, 1860, series 4, 

 vol. xx, pp. 337 — 341. On the number of trees with a given number of nodes, 

 see the Quarterly Journal of Mathematics, London, 1889, vol. urn, pp. 376 — 378. 

 The connection with chemistry was first pointed out in Cayley's paper on 

 isomers, Philosophical Magazine, June, 1874, series 4, vol. xlvii, pp. 444 — 447, 

 and was treated more fully in his report on trees to the British Association in 

 1875, Reports, pp. 257—305. 



