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In the theory of Young diagrams, it's a common move to turn the young diagram 45 degrees and consider it's profile as a function. [ Note for editors: it would be nice to find an image of this... ]

Are there ways to build 1-d (probably piecewise linear) curves from trees or from binary trees or ternary trees?

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Yes, if the tree is rooted and planar (that is, children at each vertex are linearly ordered), you can construct its contour function. See, for example, http://www.math.ens.fr/~legall/Rio-lectures.pdf (see p.2).

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