Find an optimal binary tree for a$1,$ a$2,...,$ a$6$ by using the dynamic programming method, if the identifiers a$1,$ a$2,...,$ a$6$ have probabilities $0\mathrm{.}15,0\mathrm{.}1,0\mathrm{.}2,0\mathrm{.}05,0\mathrm{.}2,0\mathrm{.}3$ respectively and all other identifiers have zero probability.