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 Tree, adding an edge creates at least one cycle.

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

Let these two cycles be $$\left({u, v, \ldots, u_1, u_2, \ldots, u}\right)$$ and $$\left({u, v, \ldots, v_1, v_2, \ldots, u}\right)$$.

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

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

From Paths in Trees are Unique, that means $$T$$ can not be a tree.

Hence the result.