Trees and forests
A tree is a connected undirected graph with no simple circuits.
A forest is an undirected graph with no simple circuits.
Theorem 1. An undirected graph is a tree iff there is a unique simple
path between any two of its vertices.
Proof: (IF) Assume there is a unique simple path between any 2 vertices.
The existence of a path between any 2 vertices implies connectivity.
And the uniqueness of every such path implies absence of simple circuits.
Indeed, if there was a simple circuit, there would be 2 paths between any
two vertices of the circuit.
(ONLY IF) - left as an exercise.