$7\mathrm{.}8$ Find an optimal binary tree for $a_1,a_2,$dots,$a_6,$ if the identifiers, in

order, have probabilities $0\mathrm{.}2,0\mathrm{.}1,0\mathrm{.}15,0\mathrm{.}2,0\mathrm{.}3,0\mathrm{.}05$ respectively

and all other identifiers have zero probability.