From the Super Mario Wiki
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Luigi is pixelated to the Hollow Tree instead of the Haunted Towers Entrance for this mission. Luigi automatically inserts the key into the door. He goes into the Eerie Staircase. Luigi has to watch the torches on the Eerie Staircase: choosing the wrong one makes Luigi slide all the way back to the bottom, with a Greenie or other ghost to defeat. Luigi has to reach to the top of the staircase to reach the Tree House. Prior to entering the Tree House, Luigi has to pull a cord to reveal a ladder.
In the Tree House, the Harsh Possessor takes control of the staircase leading to the Dark Moon piece. Luigi has to wait and avoid the staircase's attacks until it jumps, slams into the ground, and reveals the cord. Pulling the cord lets loose the Harsh Possessor, in which Luigi can suck in. Luigi has to repeat this two more times and the Dark Moon piece is his.
We've finally got our mitts on that elusive special key! Now we've just got to get to the treetop and grab the Dark Moon piece.
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