Problem $4.$ The complete, balanced $4-$ary tree of depth d$,$ denoted T$($d$),$ is defined as follows. T$0)$ consists of a single vertex. Ford $>0,$ T$($d$)$ is obtained by starting with a single vertex and setting each of its four children to be copies of T$($d $-1).$ Prove by induction that T$($d$)$ has $4$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. $($Note: T$($d$)$ is a tree and not the number of leaves on the tree. Avoid writing T$($d$)=4$d$,$ as these data types are incomparable: a tree is not a number.$]$Example $1.$ We have the following: T$(0)$ T$(1)$ T$(2)$