Prove that a graph is a tree if and only if it has no cycles and the insertion of any new edge always creates exactly one cycle.
The Attempt at a Solution
Assume that a graph G is connected and contains no vertices with a degree of zero.
So would I get my proof by proving that the graph is a tree and it is connected and has n-1 edges which proves it has no cycles and then prove that adding a edge would create a cycle making the first statement false. Does this work?