Equivalent Definitions for Finite Tree

Theorem
Let $$T$$ be a finite tree of order $$n$$.

The following statements are equivalent:


 * $$1$$: $$T$$ is connected and has no circuits.


 * $$2$$: $$T$$ has $$n-1$$ edges and has no circuits.


 * $$3$$: $$T$$ is connected and has $$n-1$$ edges.


 * $$4$$: $$T$$ is connected, and the removal of one edges renders $$T$$ is disconnected.


 * $$5$$: Any two vertices of $$T$$ are connected by exactly one path.


 * $$6$$: $$T$$ has no circuits, but adding one edge creates a circuit.

Proof
Statement $$1$$ is the usual definition of a tree.

1 implies 2

 * The fact that $$T$$ has $$n-1$$ edges is proved in Number of Edges in Tree.


 * The fact that $$T$$ has no circuits is part of statement $$1$$.

So $$1 \implies 2$$.

2 implies 3

 * The fact that $$T$$ has $$n-1$$ edges is part of statement $$2$$.

6 implies 1
Suppose $$T$$ has no circuits, but adding one edge creates a circuit.


 * The fact that $$T$$ has no circuits is part of statement $$6$$.


 * If $$T$$ were disconnected, then it would be possible to add an edge $$e$$ to connect two components of $$T$$.

By definition, $$e$$ would be a bridge.

From Condition for an Edge to be a Bridge, it follows that $$e$$ does not lie on a circuit.

So, if the only way to add an edge to $$T$$ forms a circuit, it follows that $$T$$ must be connected.

So $$T$$ is connected and has no circuits, and $$6 \implies 1$$.

Thus, all the above can be used as a definition for a finite tree.