Math 2.0

Combinatorics Algebraic combinatorics Theorem

Cayley's formula

Builds on Tree

Statement

Let n≥2n \ge 2 be an integer. Then the number of trees with vertex set {1,2,…,n}\{1, 2, \dots, n\} is nn−2n^{n-2}.

Remarks

The trees are labelled: two trees are the same only if they have the same edge set. Thus the formula counts the spanning trees of KnK_n. For n=3n = 3 there are 33 trees, and for n=4n = 4 there are 1616. A proof uses the Prüfer sequence, a bijection between these trees and the sequences of length n−2n - 2 over {1,…,n}\{1, \dots, n\}.

History

Carl Wilhelm Borchardt found the formula in 1860. Arthur Cayley extended it in 1889, and the formula has his name.

Reading path

To understand this page from the ground up, read these 10 pages in order. The foundations come first.

Show the path as a graph
Graph, definition, no evidence levelGraphdefinitionWalk, definition, no evidence levelWalkdefinitionReachability, definition, no evidence levelReachabilitydefinitionTrail, definition, no evidence levelTraildefinitionConnected graph, definition, no evidence levelConnected graphdefinitionCircuit, definition, no evidence levelCircuitdefinitionCycle, definition, no evidence levelCycledefinitionAcyclic graph, definition, no evidence levelAcyclic graphdefinitionTree, definition, no evidence levelTreedefinitionCayley's formula, statement, no evidence levelCayley's formulastatement
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Graph definition
  2. Walk definition
  3. Reachability definition
  4. Trail definition
  5. Connected graph definition
  6. Circuit definition
  7. Cycle definition
  8. Acyclic graph definition
  9. Tree definition
  10. Cayley's formula statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Cayley's_formula, lead section
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