Here is five trees that I remember from the KDramas I have seen. Since my memory tends to be focused on short-term things more than on long-term things, and trees in dramas are usually not all that well-represented and noteworthy, you shouldn't be surprised to see two very recent dramas (with relatively little tree exposure) being considered.
No 1: IljimaeIljimae was a rating hit back in 2008 and it had two of my favorite actors in it: Park Shi-hoo (how many days before I can see you again?!) and Lee Jun-ki. Ah, Iljimae ... thinking about you makes me nostalgic. And always sad. I'm still not sure whether he actually dies in the end? Or whether the ending seems to suggest he is still alive? Or is it just the idea of him that's alive?! I want to believe he lived, but am somewhat certain he died ... hence the sadness.
"branch of plum tree", which is also Iljimae's trade mark. Young Iljimae witnesses the brutal murder of his father under the blooming trees, and he meets his one true love under the same tree, and he goes and sits in the tree, and meets his one true love again in the tree, and they kiss in the tree, he makes the evil king kneel in front of the tree, etc. etc. etc. This tree actually does play a rather substantial part in this Drama, so that's why it gets to go first.
No 2: Arang and the MagistrateHow wonderful, another Lee Jun-ki drama! There's three trees I remember. No 1, the incredibly beautiful red tree in the episode when Arang goes to the Underworld. Yeah, okay, this is just scenery. But it's the most beautiful red tree I've ever seen in a drama. I think (yeah, cause my memory is not all that good, really).
MC Sniper's incredibly cool OST-track). Particularly memorable (with a touch of sadness) is that she says she wants to be reborn as a flower when she dies; or rather, a butterfly. When he grumbles that he must have seen these flowers many times before, being a ghost in this town for three years, she replies that, yes, of course, she has seen them many times already. But: "It’s the first time with you. Being with you makes the same landscape look different". Hell, girl, after this, you try to convince him you are not at all interested in him?!
Tree No 3 is the tree he falls off from in episode 7. It's the most evil tree in the show, probably because it came in touch with the evil amulet.