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

Wallner, Michael

doi:10.1016/j.tcs.2018.06.053

Cited by 3 publications

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“…In this section we give an alternative recurrence with only one auxiliary parameter (instead of two) other than the size n, which leads to an algorithm of arithmetic complexity O(n 2 ) to compute the first n terms of the sequence. The construction is motivated by the recent bijection [25].…”

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Compacted binary trees admit a stretched exponential

Price,

Fang,

Wallner

2019

Preprint

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A compacted binary tree is a directed acyclic graph encoding a binary tree in which common subtrees are factored and shared, such that they are represented only once. We show that the number of compacted binary trees of size n grows asymptotically likewhere a 1 ≈ −2.338 is the largest root of the Airy function. Our method involves a new two parameter recurrence which yields an algorithm of quadratic arithmetic complexity. We use empirical methods to estimate the values of all terms defined by the recurrence, then we prove by induction that these estimates are sufficiently accurate for large n to determine the asymptotic form. Our results also lead to new bounds on the number of minimal finite automata recognizing a finite language on a binary alphabet. As a consequence, these also exhibit a stretched exponential.

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mentioning

confidence: 99%

Compacted binary trees admit a stretched exponential

Price,

Fang,

Wallner

2019

Preprint

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Asymptotic enumeration of compacted binary trees of bounded right height

Genitrini

Gittenberger

Kauers

et al. 2020

Journal of Combinatorial Theory, Series A

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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.

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