At the end of Return of the King it implies that Legolas built his ship on the day of Aragorn's death and sails west, but to me that just doesn't work. I think one of two things must have happened either he was already building it or started building it on March 1st and left when it was completed. Its also implied he took his people with him so it must have been a reasonable sized Vessel and now some kind of rowboat. My favourite theory is that he was already building the vessel along with this people, ready for when Aragorn died. What to people think?
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I don't think it took Legolas only one day to build the ship. He began building it after Aragorn's death and finished it before the year was over. According to this site, it takes a year or two to build a ship, but these are elves building this one, and it's more than likely it had been planned out beforehand. It's possible that it had already been in the process of construction already, and Legolas put in the finishing touches.
That certainly makes logical sense.