The complete, balanced $3-$ary tree of depth d$,$ denoted T $($d$),$ is defined as follows.

$$ T $(0)$ consists of a single vertex.

$$ For d $>0,$ T $($d$)$ is obtained by starting with a single vertex and setting each of its three children to be copies of T$($d$1).$

Prove by induction that T $($d$)$ has $3$d leaf nodes. To help clarify the definition of T $($d$),$ illustrations of T $(0),$ T $(1),$ and T $(2)$ are on the next page.