Spanning Trees
Subtrees that connect all vertices.
Key idea
Subtrees that connect all vertices.
In depth
A spanning tree of a connected graph includes all vertices and enough edges to connect them without cycles. A connected graph has at least one, and the number can be counted by the matrix tree theorem.
Worked example
K₃ (a triangle) has 3 spanning trees, each omitting one edge.
Real-world uses
- Designing a network of cables that connects all buildings with minimal wire
Common mistake
Forgetting that a spanning tree must include every vertex.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Spanning Trees?
Subtrees that connect all vertices. It is a topic at the advanced level in Discrete Math & Number Theory.
Where is Spanning Trees used in real life?
Real-world examples: Designing a network of cables that connects all buildings with minimal wire.
What should I learn before Spanning Trees?
Make sure you are comfortable with Trees, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Spanning Trees?
Forgetting that a spanning tree must include every vertex.
How can I practise Spanning Trees?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Spanning Trees?
Move on to Minimum Spanning Trees next.
Related advanced topics in Discrete Math & Number Theory
- Tautologies and Contradictions – Always true or always false statements.
- Logical Equivalence – Statements with identical truth tables.
- De Morgan’s Laws – Negation swaps AND and OR.
- Converse and Contrapositive – A conditional equals its contrapositive.
- Quantifiers – “For all” and “there exists”.
- Negating Quantified Statements – Flip the quantifier and negate the predicate.