Math 2.0

Combinatorics Theorem

A tree on n vertices has n − 1 edges

Builds on Edge setTree

Statement

Let TT be a finite tree with nn vertices. Then TT has exactly n−1n - 1 edges.

Remarks

A tree has at least one vertex, because it is connected. Thus n≥1n \ge 1. Proof by induction on nn: for n≥2n \ge 2 delete a leaf, which exists by the leaf lemma. The rest is a tree with n−1n - 1 vertices. A consequence: the sum of the degrees in a tree is 2n−22n - 2.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
Graph, definition, no evidence levelGraphdefinitionEdge set, definition, no evidence levelEdge setdefinitionWalk, 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 levelTreedefinitionA tree on n vertices has n − 1 edges, statement, no evidence levelA tree on n vertices …statement
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. Edge set definition
  3. Walk definition
  4. Reachability definition
  5. Trail definition
  6. Connected graph definition
  7. Circuit definition
  8. Cycle definition
  9. Acyclic graph definition
  10. Tree definition
  11. A tree on n vertices has n − 1 edges statement

Further reading

  1. Web https://en.wikipedia.org/wiki/Tree_(graph_theory), Properties
About this page
Tag0VSNHTHK8T
URIm2://0VSNHTHK8T
Kindstatement, theorem
WritingW0 stub
MSC05C05
StatusThis copy of the site has no build data. Run m2 status 0VSNHTHK8T.
UsesEdge set, Tree

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