how to understand it
What does "lip-perty-lipperty-lip" means? The whole sentence is "Peter at once took himself off to the Green Forest, lip-perty-lipperty-lip, as fast as he could go."

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I assume Peter is some type of animal, in which case it probably reflects the sound his feet make on the ground when he runs.

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It really doesn't mean anything.  The author is basically just trying to make the story more fun for children.  A parent who is reading this to his or her children would sound funny to the children.