Networks and Spanning Trees

In 1857 Arthur Cayley (1821–1895) published a paper [9] that introduces the term “tree” to describe the logical branching that occurs when iterating the fundamental process of (partial) differentiation. When discussing the composition of four symbols that involve derivatives, Cayley writes “But without a more convenient notation, it would be difficult to find [their] corresponding expressions . . . . This, however, can be at once effected by means of the analytical forms called trees . . . ” [9]. Without defining the term “tree,” Cayley has identified a certain structure that occurs today in quite different situations, from networks in computer science to representing efficient delivery routes for transportation. In the paper “On the Theory of the Analytical Forms Called Trees” [9], Cayley is intrigued enough by this new structure that he proceeds to count trees with certain properties. For him, every tree represents a sequence of derivatives applied in a very specific order, terminating at a final or root term denoted U . Cayley actually uses the word “root” in reference to the point corresponding to U . The remainder of the paper enumerates what today are called “rooted trees.” However, in a later paper “A Theorem on Trees” [10], published in 1889, Cayley makes a finer distinction when counting trees, so that no one point is considered as the root, but all points carry fixed labels α, β, γ, etc. The British mathematician counts these trees with fixed labels, arriving at a result that today is called “Cayley’s formula.” Cayley associates to each labeled tree a polynomial, and proceeds to add all polynomials corresponding to labeled trees with n vertices, arriving at a compelling result that depends on n in a very recognizable pattern. The reader is invited to find this pattern, and perhaps follow in Cayley’s footsteps of discovery, by working Exercises (2.4), (2.5), (2.6). After systematically counting labeled trees on six vertices, Cayley writes: “It will be at once seen that the proof for this particular case is applicable for any value whatever of n” [10]. This “proof,” however, would require an inverse correspondence between Cayley polynomials and labeled trees, which he does not construct. In fact, most of Cayley’s polynomials correspond to several possible trees, as outlined in Exercise (2.8). A complete proof of Cayley’s result is offered from the work of the German mathematician Heinz Prüfer (1896–1934), who develops a quite clever and geometrically appealing method for counting labeled trees. He uses no modern terminology, not even the word “tree” in his work. Probably at the suggestion of Issai Schur (1875–1941), Prüfer phrases his arguments in terms of counting railway networks with certain properties [21]: Given a country with n towns, in how many ways can a railway network be constructed so that