On Succinct Representations of Binary Trees

Abstract

We observe that a standard transformation between ordinal trees (arbitrary rooted trees with ordered children) and binary trees leads to interesting succinct binary tree representations. There are four symmetric versions of these transformations. Via these transformations we get four succinct representations of n-node binary trees that use $2n + n/(\log n)^{\Theta (1)}$ bits and support (among other operations) navigation, inorder numbering, one of preorder or postorder numbering, subtree size and lowest common ancestor queries. While this functionality, and more, is also supported in O(1) time using $2n + o(n)$ bits by Davoodi et al. (Phil Trans R Soc A 372:20130131, 2014) extension of a representation by Farzan and Munro (Algorithmica 6), their redundancy, or the o(n) term, is much larger, and their approach may not be suitable for practical implementations. One of these transformations is related to the Zaks’ sequence (Theor Comput Sci 10:63–82, 1980) for encoding binary trees, and we thus provide the first succinct binary tree representation based on Zaks’ sequence. The ability to support inorder numbering is crucial for the well-known range-minimum query problem on an array A of n ordered values. Another of these transformations is equivalent to Fischer and Heun’s (SIAM J Comput 40(2):465–492, 2011) 2d-Min-Heap structure for this problem. Yet another variant allows an encoding of the Cartesian tree of A to be constructed from A using only $O(\sqrt{n} \log n)$ bits of working space.