Summary of Graph Theory 11: Trees

Main Ideas and Concepts:

• Definition of a Tree:
  • A Tree is defined as a connected graph that contains no cycles.
  • A cycle is described as a collection of distinct Edges and Vertices that form a closed loop.
• Importance of Trees:
  • Trees are significant in data storage and representation, exemplified by Family Trees where relationships can be visualized hierarchically.
• Properties of Trees:
  • Leaves: Trees contain Vertices with a degree of exactly one, known as Leaves.
  • Unique Paths: Any two Vertices in a Tree are connected by exactly one unique path, unlike in graphs with cycles where multiple paths may exist between Vertices.
  • Edge Count: For a Tree with n Vertices, there are n - 1 Edges. This property can be verified through a specific example and mathematical induction.

Methodology for Proving Properties:

• Induction Proof for Edge Count:
  • Base Case: For n = 2 (two Vertices), there is one edge connecting them.
  • Inductive Step: Assume true for K Vertices. For K + 1 Vertices:
    • Remove a leaf (vertex with degree 1) and its connecting edge from the Tree.
    • This results in a Tree with K Vertices, maintaining the property that the number of Edges is one less than the number of Vertices.
    • Conclude that the original Tree had K + 1 Vertices and K Edges, thus confirming the relationship n - 1.

Equivalent Definitions of Trees:

• A Tree can be defined in various equivalent ways:
  • As a connected graph without cycles.
  • As a graph where every two Vertices are connected by unique paths.
  • As a graph with n Vertices and n - 1 Edges.

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