Trees
Connected graphs without cycles; n vertices have n−1 edges.
Key idea
Connected graphs without cycles; n vertices have n−1 edges.
In depth
A tree is a connected graph with no cycles. A tree on n vertices has exactly n − 1 edges, and a unique path between any two vertices. Removing any edge disconnects it.
Worked example
A tree with 10 vertices has 9 edges.
Formula
Real-world uses
- Organising file folders, family trees and decision-making processes
- The structure of many data structures
Common mistake
Believing a connected graph with n − 1 edges and a cycle could exist; it cannot.
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 Trees?
Connected graphs without cycles; n vertices have n−1 edges. It is a topic at the advanced level in Discrete Math & Number Theory.
What is the formula for Trees?
The key formula is |E| = |V| − 1.
Where is Trees used in real life?
Real-world examples: Organising file folders, family trees and decision-making processes; The structure of many data structures.
What should I learn before Trees?
Make sure you are comfortable with Hamiltonian Paths, the previous topic in Discrete Math & Number Theory.
What is a common mistake with Trees?
Believing a connected graph with n − 1 edges and a cycle could exist; it cannot.
How can I practise 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 Trees?
Move on to 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.