[She paid attention as he was writing, stubbornly trying to guess Aim's name from the pen's movement, but static vision flashed over her gaze for a split second. The next thing she knew Gene's writing had already turned to inkblots. Apparently the Tree was more stubborn than she was.]

Oh. So even thrice is possible...

[She waited for him to finish writing, then flipped to the next leaf, containing the note she wrote after the Frost Mountain trip: The entrance was free, to exit has a cost. Answer the riddle and no more will you be lost. "What has roots as nobody sees, Is taller than trees, Up, up it goes, And yet never grows?" = MOUNTAIN]

A carving on the cave wall. We figured out the answer and as sudden as the entrance vanished, we found ourselves in the dojo.