Math 2.0

statement theorem

Unique paths in trees

Statement

Let GG be a graph. Then GG is a tree if and only if GG has at least one vertex and these conditions hold. For any two vertices uu and vv of GG, there is exactly one path from uu to vv.

Remarks

Existence of a path between any two vertices is connectedness. Uniqueness of the path is equivalent to acyclicity: two different paths between the same vertices contain a cycle. In a tree, the unique path from uu to vv is a shortest walk, so its length is the distance d(u,v)d(u, v).

Explanations

L1 human

In a tree there is only one way to get from one vertex to another without going back. If there were two different ways, together they would form a loop, and a tree has no loop.

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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 levelGraphdefinitionWalk, definition, no evidence levelWalkdefinitionPath, definition, no evidence levelPathdefinitionReachability, 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 levelTreedefinitionUnique paths in trees, statement, no evidence levelUnique paths in treesstatement
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. Walk definition
  3. Path 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. Unique paths in trees statement

Sources

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

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