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Enumeration of Ordered Trees with Leaf Restrictions

Authors Yasuaki Kobayashi , Dominik Köppl , Yasuko Matsui , Hirotaka Ono , Toshiki Saitoh , Yushi Uno

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Enumeration
  • Mathematics of computing → Trees
  • Theory of computation
Keywords
  • binary trees
  • ordered trees
  • rooted trees
  • enumeration algorithm
  • constant-time delay

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Abstract

An α-ary tree for a constant α ≥ 2 is a rooted tree in which each node has at most α children. A node having no children is called a leaf. For a given rooted tree and a node v, the number of edges from the root to v is called the depth of v. We call a vector w = (w_1,w_2,…, w_d) of nonnegative integers an (α-ary) distribution if there is an α-ary tree T such that the number of leaves at each depth i ∈ [1..d] in T is w_i. Although not every vector of nonnegative integers is a distribution, a distribution can be associated with many α-ary trees. In this paper, we present an algorithm to enumerate all α-ary trees for a given distribution. Our algorithm reports the first tree in O(d + ∑_{i = 1}^d w_i) time, and then each subsequent α-ary tree in O(max_{i = 1}^d w_i) time by representing each tree as the difference from the previous one. The algorithm can be restricted to computing all trees that are full, i.e., trees whose nodes have exactly α or no children.

Cite As Get BibTex

Yasuaki Kobayashi, Dominik Köppl, Yasuko Matsui, Hirotaka Ono, Toshiki Saitoh, and Yushi Uno. Enumeration of Ordered Trees with Leaf Restrictions. In From Strings to Graphs, and Back Again: A Festschrift for Roberto Grossi's 60th Birthday. Open Access Series in Informatics (OASIcs), Volume 132, pp. 8:1-8:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/OASIcs.Grossi.8

Author Details

Yasuaki Kobayashi
  • Faculty of Information Science and Technology, Hokkaido University, Sapporo, Japan
Dominik Köppl
  • Department of Informatics, Yamanashi University, Japan
Yasuko Matsui
  • Department of Mathematical Sciences, Tokai University, Japan
Hirotaka Ono
  • Department of Mathematical Informatics, Nagoya University, Japan
Toshiki Saitoh
  • School of Computer Science and Systems Engineering, Kyushu Institute of Technology, Fukuoka, Japan
Yushi Uno
  • Graduate School of Informatics, Osaka Metropolitan University, Japan

Funding

  • Kobayashi, Yasuaki: JSPS KAKENHI Grant Numbers JP23K28034, JP24H00686, and JP24H00697.
  • Köppl, Dominik: JSPS KAKENHI Grant Numbers JP23H04378 and JP25K21150.
  • Matsui, Yasuko: JSPS KAKENHI Grant Numbers JP20H05964 and JP20K04973.
  • Ono, Hirotaka: JSPS KAKENHI Grant Numbers JP20H05967, JP22H00513, JP24K02898, and JP25K03077, and JST CRONOS Grant Number JPMJCS24K2.
  • Saitoh, Toshiki: JSPS KAKENHI Grant Number JP21H05857.
  • Uno, Yushi: JSPS KAKENHI Grant Numbers JP20H05964 and JP21K11757.

References

  1. Masahiro Baba, Hirotaka Ono, Kunihiko Sadakane, and Masafumi Yamashita. A succinct representation of a full binary tree. Technical Report 0, Record of Joint Conference of Electrical and Electronics Engineers in Kyushu, 2009. URL: https://doi.org/10.11527/jceeek.2009.0.59.0.
  2. David Benoit, Erik D. Demaine, J. Ian Munro, Rajeev Raman, Venkatesh Raman, and S. Srinivasa Rao. Representing trees of higher degree. Algorithmica, 43(4):275-292, 2005. URL: https://doi.org/10.1007/s00453-004-1146-6.
  3. Terry Beyer and Sandra Mitchell Hedetniemi. Constant time generation of rooted trees. SIAM J. Comput., 9(4):706-712, 1980. URL: https://doi.org/10.1137/0209055.
  4. Michael Buro. On the maximum length of Huffman codes. Inf. Process. Lett., 45(5):219-223, 1993. URL: https://doi.org/10.1016/0020-0190(93)90207-P.
  5. Jeremy Chizewer, Stephen Melczer, J. Ian Munro, and Ava Pun. Enumeration and succinct encoding of AVL trees. In Proc. AofA, volume 302 of LIPIcs, pages 2:1-2:12, 2024. URL: https://doi.org/10.4230/LIPICS.AOFA.2024.2.
  6. Nachum Dershowitz and Shmuel Zaks. Enumerations of ordered trees. Discret. Math., 31(1):9-28, 1980. URL: https://doi.org/10.1016/0012-365X(80)90168-5.
  7. Sen-Peng Eu, Seunghyun Seo, and Heesung Shin. Enumerations of vertices among all rooted ordered trees with levels and degrees. Discret. Math., 340(9):2123-2129, 2017. URL: https://doi.org/10.1016/J.DISC.2017.04.007.
  8. Frank Gray. Pulse code communication. US Patent 2632058, 1953. Google Scholar
  9. Roberto Grossi. Enumeration of paths, cycles, and spanning trees. In Encyclopedia of Algorithms, pages 640-645. Springer, 2016. URL: https://doi.org/10.1007/978-1-4939-2864-4_728.
  10. Guy Jacobson. Space-efficient static trees and graphs. In Proc. FOCS, pages 549-554, 1989. URL: https://doi.org/10.1109/SFCS.1989.63533.
  11. Leon Gordon Kraft. A device for quantizing, grouping, and coding amplitude-modulated pulses. Master’s thesis, Massachusetts Institute of Technology, 1949. Google Scholar
  12. Shin-Ichi Nakano and Takeaki Uno. Constant time generation of trees with specified diameter. In Proc. WG, volume 3353 of LNCS, pages 33-45, 2004. URL: https://doi.org/10.1007/978-3-540-30559-0_3.
  13. Frank Ruskey and Andrzej Proskurowski. Generating binary trees by transpositions. J. Algorithms, 11(1):68-84, 1990. URL: https://doi.org/10.1016/0196-6774(90)90030-I.
  14. Ivan Stojmenovic. A simple systolic algorithm for generating combinations in lexicographic order. Computers & Mathematics with Applications, 24(4):61-64, 1992. Google Scholar
  15. Takeaki Uno. Two general methods to reduce delay and change of enumeration algorithms. Technical Report NII-2003-004E, NII Technical Report, 2003. URL: https://doi.org/10.20736/0000000385.
  16. Timothy R. Walsh. Loop-free sequencing of bounded integer compositions. Journal of Combinatorial Mathematics and Combinatorial Computing, 33:323-345, 2000. Google Scholar
  17. Katsuhisa Yamanaka, Yota Otachi, and Shin-Ichi Nakano. Efficient enumeration of ordered trees with k leaves. Theor. Comput. Sci., 442:22-27, 2012. URL: https://doi.org/10.1016/j.tcs.2011.01.017.
  18. Shmuel Zaks. Lexicographic generation of ordered trees. Theor. Comput. Sci., 10:63-82, 1980. URL: https://doi.org/10.1016/0304-3975(80)90073-0.
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