The Dunk and Egg stories take place around 200 AC.
The Knight of the Laughing Tree and the Tourney of Harrenhal take place around 282 (281?) AC. Dunk is long dead.
Duncan did have a tree a his sigil though. His was an Elm tree with a shooting star. He also did appear as a mystery knight once so far, but his sigil for that was a hanged man.
Speaking of sigils, Ned is the only candidate for whom the laughing tree sigil makes any sense. Lyanna is never established as pious at any point. Ned constantly prays to his Tree gods and when he was young he used to laugh with Robert a lot.
KotLT = Ned on one of Bob's "giant destriers", making him appear small (just as the Mountain makes his horse look like a pony at the other Tourney Ned attends).