Characteristics of Finite Tree

Theorem

Let $T$ be a finite simple graph of order $n$.

The following statements are equivalent:

Condition $1$

$T$ is a finite tree of order $n$ if and only if $T$ has $n - 1$ edges and has no circuits.

Condition $2$

$T$ is a finite tree of order $n$ if and only if $T$ has $n - 1$ edges and is connected.

Condition $3$

$T$ is a finite tree if and only if two arbitrary vertices of $T$ are connected by exactly one path.

Condition $4$

$T$ is a finite tree if and only if $T$ has no circuits, but adding one edge creates a cycle.