Finite Connected Graph is Tree iff Size is One Less than Order/Necessary Condition/Induction Step/Proof 2

Theorem
Let the following hold:
 * For all $j \le k$, a tree of order $j$ is of size $j-1$.

Then this holds:
 * A tree of order $k+1$ is of size $k$.

Proof
Let $T_{k+1}$ be any tree with $k+1$ nodes.

Remove any edge $e$ of $T_{k+1}$.

By definition of tree $T_{k+1}$ has no circuits

Therefore from Condition for Edge to be Bridge it follows that $e$ must be a bridge.

So removing $e$ disconnects $T_{k+1}$ into two trees $T_1$ and $T_2$, with $k_1$ and $k_2$ nodes, where $k_1 + k_2 = k+1$.

By hypothesis, $T_1$ and $T_2$ have $k_1 - 1$ and $k_2 - 1$ edges.

Putting the edge $e$ back again, it can be seen that $T_{k+1}$ has $\left({k_1 - 1}\right) + \left({k_2 - 1}\right) + 1 = k$ edges.

So $P \left({k}\right) \implies P \left({k+1}\right)$ and the result follows by the Principle of Strong Induction.

Therefore a tree of order $k+1$ is of size $k$.