Graphs / Trees
Least You Need to Know: Trees
A tree is a connected graph with **no cycles**. In a tree with n vertices, the number of edges is always n-1.
The least you need to know
- A tree is connected and acyclic.
- A graph with n vertices and n-1 edges is a tree if it is connected.
- Adding one edge to a tree creates exactly one cycle.
- Removing one edge from a tree disconnects it.
Key notation
n
number of vertices
n-1
edge count in a tree
leaf
vertex of degree 1
Tiny worked example
- A tree has 7 vertices.
- Then it has 6 edges.
- If you add one new edge between two existing vertices, you create one cycle.
Common mistakes
- Students often memorize n-1 but forget the graph also has to be connected.
- Students often call any acyclic graph a tree even if it has multiple components.
- Students often forget leaves have degree 1, not 0.
How to recognize this kind of problem
- If the graph is not connected, it is not a tree.
- Count vertices first, then compare with edges.
- Ask whether there is exactly one simple path between two vertices.
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