I did a bit of testing on it for you
So when there is 1 player it always blow towards the players side.
When there are two, 1 player on each side it blows randomly I guess.
When there are two, 2 players on one side it always blow to that side.
Also, if one player stands on the top platform in the middle he isnt counted and it blows one way. But if he teeters to either side he can be seen and it resumes the random.
The top platform also isnt effected by wind. and because the tree is slightly to the left the wind effect area is larger on the right side.
Im not sure if its a random pattern or just random. The first time I got x2 L, x2R. Second and third time it was just weird ****.
Ok this actually intrigued me, so I did some testing of my own using the single player glitch.
what I did was I stood in the dead center of the stage, and saw what happened.
What I observed was that the tree would constantly have the wind blow to the left, yet I would not move. I was playing as fox if that is relevant. Also I'm not sure if this is something original but I thought it was something that was pretty cool.
If someone could recreate this and confirm, I would appreciate it. Furthermore I'm not sure if the tree is just blowing to the left because that was the last direction it blew prior to me placing the character in the dead center and it since it is not ambiguous as to which side I am on it continues to blow in that direction.
Edit: I went into the center again after moving to the right, the tree began to blow to the left only after I walked into the middle of the stage (note when I say middle I mean on the ground not on a platform).
Edit #2: I am not sure how large this "middle ground" is, but it seems to be a couple of steps as fox, so it need not be pixel prefect (unless I have managed to place myself correctly by guesswork exactly each time).
Edit#3: Kaostar, I tested the wind when standing on the top platform, and it blew according to which side I was standing on, regardless of teetering it would switch if I crossed to either the right or the left side. That is to say if I was standing on the right side it blew to the right and vice versa for the left. (1 player mode)
Edit#4: I tested out the idea that when one player is on the top platform, and another is on one side of the stage, the wind blows in that direction, I did not find that correlation, what I found was that the player on the top platform was still counted as if he were on the ground, that is to say if he was on the right part of the platform and the other player was also on the right side of the whole stage, the wind would blow right, and vice versa for the left. If the player on the platform was on the left side and the other player is on the right side, it will blow in a random direction.
I'm also using version 1.2 if that could have any effect on DL64.
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Another thing that I "think I found" is an oddity with fox's taunting. When fox is on the edge of a platform, and is teetering, and you taunt, afterwards you will be in a standing position. Taunt again as fox and you will remain in a standing position. Taunt a third time and you will be teetering after you are finished taunting.
I can also confirm that this works for sheik, only in iterations of 4 instead of 3.
I tested this on the rest of the cast up to about 7 or so taunts, no other characters had this trait besides sheik and fox.