Math 2.0

statement theorem

A tree on n vertices has n − 1 edges

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.

Explanations

L1 human

Build a tree by adding one vertex at a time, each joined to the tree by one new edge. You start with 11 vertex and 00 edges, so at the end there is always one edge fewer than vertices.

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What this needs

To read this node, read these 11 nodes in this order. The foundations come first, and the list ends with this node.

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 node uses the nodes above it. The colour of the bar on a box shows the evidence level, and the box also gives it as text. The list below has the same nodes.
  • 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

Sources

  1. Web https://en.wikipedia.org/wiki/Tree_(graph_theory), Properties

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