Math 2.0

Combinatorics Definition

Prüfer sequence

Builds on Tree

Definition

Let n≥2n \ge 2 be an integer and let TT be a tree with vertex set {1,…,n}\{1, \dots, n\}. The Prüfer sequence of TT is the sequence (a1,…,an−2)(a_1, \dots, a_{n-2}) that this process gives. Let T0=TT_0 = T. For i=1,…,n−2i = 1, \dots, n - 2, let ℓi\ell_i be the leaf of Ti−1T_{i-1} with the smallest label. Let aia_i be the unique neighbour of ℓi\ell_i in Ti−1T_{i-1}, and let Ti=Ti−1−ℓiT_i = T_{i-1} - \ell_i.

Remarks

The map T↦(a1,…,an−2)T \mapsto (a_1, \dots, a_{n-2}) is a bijection from the labelled trees on {1,…,n}\{1, \dots, n\} to {1,…,n}n−2\{1, \dots, n\}^{n-2}. This proves Cayley's formula. A vertex vv appears deg⁡(v)−1\deg(v) - 1 times in the sequence. Thus the leaves are the labels that do not appear. Example: the star with centre 11 on {1,2,3,4}\{1, 2, 3, 4\} has the sequence (1,1)(1, 1).

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

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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 levelTreedefinitionPrüfer sequence, definition, no evidence levelPrüfer sequencedefinition
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. Prüfer sequence definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Prüfer_sequence, Algorithm to convert a tree into a Prüfer sequence
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