Cartesian tree: Difference between revisions

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[[File:Cartesian tree.svg|thumb|240px|A sequence of numbers and the Cartesian tree derived from them.]]

{{for|Descartes' metaphor of tree of knowledge|Tree of knowledge (philosophy)}}

In [[computer science]], a '''Cartesian tree''' is a [[binary tree]] derived from a sequence of distinct numbers. TheTo smallestconstruct numberthe inCartesian thetree, sequenceset isits root to atbe the rootminimum ofnumber in the tree;sequence, and recursively construct its left and right subtrees are constructed recursively from the subsequences to the leftbefore and right ofafter this number. When all numbers are distinct, the Cartesian treeIt is uniquely defined from the properties that it is [[Heap (data structure)|heap]]-ordered and that a [[Tree traversal|symmetric (in-order) traversal]] of the tree returns the original sequence.

Cartesian trees were introduced by {{harvtxt|Vuillemin|1980}} in the context of geometric [[range searching]] data structures. They have also been used in the definition of the [[treap]] and [[randomized binary search tree]] data structures for [[binary search]] problems, in [[comparison sort]] algorithms that perform efficiently on nearly-sorted inputs, and as the basis for [[pattern matching]] algorithms. A Cartesian tree for a sequence can be constructed in [[linear time]].