Abstract
One can build a Cartesian tree from an n-element sequence in O(n) time, and from an n-node free tree in O(nlogn) time (with a matching worst-case lower bound in the comparison model of computation). We connect these results together by describing a Cartesian tree construction algorithm based on a “bitonicity transform” running in O(nlogk) time on a free tree with k leaves, noting that a path is the special case of a tree with just 2 leaves. We also provide a matching worst-case lower bound in the comparison model.
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