Adding Edge to Tree Creates One Cycle

Theorem
Adding a new edge to a tree can create no more than one cycle.

Proof
From Equivalent Definitions for Finite Tree, adding an edge creates at least one cycle.

Suppose that adding an edge $\tuple {u, v}$ to a tree $T$ creates two or more cycles.

Let two such cycles be $\tuple {u, v, \ldots, u_1, u_2, \ldots, u}$ and $\tuple {u, v, \ldots, v_1, v_2, \ldots, u}$.

By removing the edge $\tuple {u, v}$ from this cycle, we have two paths from $v$ to $u$:
 * $\tuple {v, \ldots, u_1, u_2, \ldots, u}$
 * $\tuple {v, \ldots, v_1, v_2, \ldots, u}$.

But that means $T$ has more than one path between two nodes.

From Path in Tree is Unique, that means $T$ can not be a tree.

Hence the result.