Asymptotic enumeration of compacted binary trees of bounded right height

Abstract

A compacted tree is a graph created from a binary tree such that repeatedly occurring subtrees in the original tree are represented by pointers to existing ones, and hence every subtree is unique. Such representations form a special class of directed acyclic graphs. We are interested in the asymptotic number of compacted trees of given size, where the size of a compacted tree is given by the number of its internal nodes. Due to its superexponential growth this problem poses many difficulties. Therefore we restrict our investigations to compacted trees of bounded right height, which is the maximal number of edges going to the right on any path from the root to a leaf.We solve the asymptotic counting problem for this class as well as a closely related, further simplified class.For this purpose, we develop a calculus on exponential generating functions for compacted trees of bounded right height and for relaxed trees of bounded right height, which differ from compacted trees by dropping the above described uniqueness condition. This enables us to derive a recursively defined sequence of differential equations for the exponential generating functions. The coefficients can then be determined by performing a singularity analysis of the solutions of these differential equations.Our main results are the computation of the asymptotic numbers of relaxed as well as compacted trees of bounded right height and given size, when the size tends to infinity. Proposition 4.3. From any binary tree of size n, we can build a compacted tree of size n, with the following operations:1. Add a leaf as left child of the leftmost node of the binary tree.2. Add pointers to every node such that every node except the leaf has out-degree 2.

Strict monotonic trees arising from evolutionary processes: Combinatorial and probabilistic study

“…This work is a part of a long-term over-arching project, in which we aim at relaxing the classical rules of increasing labelling (described in, e.g., [BFS92]), by, for example, allowing labels to appear more than once in the tree. The following papers are part of this strand of research: [BGGW20,BGNS20] introduce and study models of label trees with less-constrained increasing labelling rules, but also other graphs structures like [BDF + 16] focuses on increasingly-labelled "diamonds" and [GGKW20] at a compacted structure that specifies classes of directed acyclic graphs.…”

Families of Monotonic Trees: Combinatorial Enumeration and Asymptotics

“…Acknowledgment. We thank Stephan Wagner for a fruitful discussion about the relationship of an involved proof of this paper and the article [8]. Furthermore we are grateful for the anonymous reviewers whose comments and suggestions helped improving and clarifying this manuscript.…”

A bijection of plane increasing trees with relaxed binary trees of right height at most one

“…The gain in memory was studied by Flajolet, Sipala, and Steyaert in [9]. The corresponding procedure defines a subclass of DAGs, called compacted trees, which are in bijection with the original trees, see [9,10]. The characterizing property of the generated structure is the uniqueness of each subtree which in the end brings savings in memory.…”

“…The level of a node is the number of right edges on the path from the root to this node, see Figure 3. The asymptotic counting problem for relaxed (and the more restrictive class of compacted) binary trees when restricted to being of finite right height was solved in [10]. Figure 4: The structure of a relaxed binary tree with right height at most one.…”

“…The general structure of relaxed binary trees of right height at most one is shown in Figure 4. In [10,Theorem 7.3] it was shown that they admit the exponential generating function…”