The board may direct the city forester to let contracts for all or any part of any improvement ordered by the city council. The city forester shall let the said contracts to the lowest responsible bidder, after notice by publication in two successive issues of any daily, or in one issue of any weekly, newspaper published and circulated in the city, or if there is no such newspaper, by posting for five days at three public places in the city. The work so contracted for shall be done under the supervision of the city forester.