Characteristics of Finite Tree
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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.