Abstract
An order relation on {1,..., n} is called tree-definable if it can be defined as the sequence of leaf numbers in a depth-first traversal of a binary tree, with leaves numbered 1,..., n, from left to right. Several characterisations of this class are given, one of them in terms of avoided patterns. Closure properties and both computational and descriptional complexity of the class are investigated. The complexity seems to depend on the representation: for the representation by a sequence, tree-definability can be described in transitive closure logic, or in monadicΠ 11 , and an algorithm is given which solves the problem in LOGDCFL. For order relations tree-definability can be expressed in first-order logic, and is therefore in A C0.
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Lautemann, C. Tree-Definable Linear Orders. Order 15, 119–128 (1998). https://doi.org/10.1023/A:1006165114181
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DOI: https://doi.org/10.1023/A:1006165114181