Abstract A fringe subtree of a rooted tree is a subtree induced by one of the vertices and all its descendants. We consider the problem of estimating the number of distinct fringe subtrees in random trees under a generalized notion of distinctness, which allows for many different interpretations of what “distinct” trees are. The random tree models considered are simply generated trees and families of increasing trees (recursive trees, d-ary increasing trees and generalized plane-oriented recursive trees). We prove that the order of magnitude of the number of distinct fringe subtrees (under rather mild assumptions on what ‘distinct’ means) in random trees with n vertices is $$n/\sqrt{\log n}$$ n / log n for simply generated trees and $$n/\log n$$ n / log n for increasing trees.

We present a new universal source code for distributions of unlabeled binary and ordinal trees that achieves optimal compression to within lower order terms for all tree sources covered by existing universal codes. At the same time, it supports answering many navigational queries on the compressed representation in constant time on the word-RAM; this is not known to be possible for any existing tree compression method. The resulting data structures, "hypersuccinct trees", hence combine the compression achieved by the best known universal codes with the operation support of the best succinct tree data structures. We apply hypersuccinct trees to obtain a universal compressed data structure for range-minimum queries. It has constant query time and the optimal worst-case space usage of $2n+o(n)$ bits, but the space drops to $1.736n + o(n)$ bits on average for random permutations of $n$ elements, and $2\lg\binom nr + o(n)$ for arrays with $r$ increasing runs, respectively. Both results are optimal; the former answers an open problem of Davoodi et al. (2014) and Golin et al. (2016). Compared to prior work on succinct data structures, we do not have to tailor our data structure to specific applications; hypersuccinct trees automatically adapt to the trees at hand. We show that they simultaneously achieve the optimal space usage to within lower order terms for a wide range of distributions over tree shapes, including: binary search trees (BSTs) generated by insertions in random order / Cartesian trees of random arrays, random fringe-balanced BSTs, binary trees with a given number of binary/unary/leaf nodes, random binary tries generated from memoryless sources, full binary trees, unary paths, as well as uniformly chosen weight-balanced BSTs, AVL trees, and left-leaning red-black trees.

Baier (CPM 2018) describes tunneling as a technique to further exploit redundancies in the Burrows-Wheeler Transform. In this paper we show how to retain indexed text searching on the resulting structure and generalize the concept to Wheeler graphs.

The definition of kth-order empirical entropy of strings is extended to node-labeled binary trees. A suitable binary encoding of tree straight-line programs (that have been used for grammar-based tree compression before) is shown to yield binary tree encodings of size bounded by the kth-order empirical entropy plus some lower order terms. This generalizes recent results for grammar-based string compression to grammar-based tree compression. A long version of this paper can be found in [11].

The problem of universal source coding for binary trees is considered. Zhang, Yang, and Kieffer derived upper bounds on the average-case redundancy of codes based on directed acyclic graph (DAG) compression for binary tree sources with certain properties. In this paper, a natural class of binary tree sources is presented such that the demanded properties are fulfilled. Moreover, for both subclasses considered in the paper of Zhang, Yang, and Kieffer, their result is improved by deriving bounds on the maximal pointwise redundancy (or worst-case redundancy) instead of the average-case redundancy. Finally, using context-free tree grammars instead of DAGs, upper bounds on the maximal pointwise redundancy for certain binary tree sources are derived. This yields universal codes for new classes of binary tree sources.

We study the average size of the minimal directed acyclic graph (DAG) with respect to so-called leaf-centric binary tree sources as studied by Zhang, Yang, and Kieffer. A leaf-centric binary tree source induces for every n >= 2 a probability distribution on all binary trees with n leaves. We generalize a result shown by Flajolet, Gourdon, Martinez and Devroye according to which the average size of the minimal DAG of a binary tree that is produced by the binary search tree model is Theta(n / log n).