Suppose we have an optimal prefix code on a set $C=\{0,1,$dots,$n-1\}$ of characters and we wish to transmit this code using as few bits as possible. Show how to represent any optimal prefix code on $C$ using only $2n-1+n|$~$logn$~$|$ bits. $($Hint: Use $2n-1$ bits to specify the structure of the tree, as discovered by a walk of the tree.$)$