How to Fit a Tree in a Box

  • Hugo A. Akitaya
  • Maarten Löffler
  • Irene ParadaEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11282)


We study compact straight-line embeddings of trees. We show that perfect binary trees can be embedded optimally: a tree with \(n\) nodes can be drawn on a \(\sqrt{n}\) by \(\sqrt{n}\) grid. We also show that testing whether a given binary tree has an upward embedding with a given combinatorial embedding in a given grid is NP-hard.



This research was initiated at the 33rd Bellairs Winter Workshop on Computational Geometry in 2018. We would like to thank all participants of the workshop for fruitful discussions on the topic. H.A.A. was supported by NSF awards CCF-1422311 and CCF-1423615, and the Science Without Borders scholarship program. M.L. was partially supported by the Netherlands Organisation for Scientific Research (NWO) through grant number 614.001.504. I.P. was supported by the Austrian Science Fund (FWF) grant W1230.


  1. 1.
    Akitaya, H.A., Löffler, M., Parada, I.: How to fit a tree in a box. CoRR (2018).
  2. 2.
    Bachmaier, C., Matzeder, M.: Drawing unordered trees on k-grids. In: Rahman, M.S., Nakano, S. (eds.) WALCOM 2012. LNCS, vol. 7157, pp. 198–210. Springer, Heidelberg (2012). Scholar
  3. 3.
    Biedl, T.: Ideal drawings of rooted trees with approximately optimal width. J. Graph Algorithms Appl. 21(4), 631–648 (2017). Scholar
  4. 4.
    Biedl, T., Mondal, D.: On upward drawings of trees on a given grid. In: Frati, F., Ma, K.-L. (eds.) GD 2017. LNCS, vol. 10692, pp. 318–325. Springer, Cham (2018). Scholar
  5. 5.
    Chan, T.M.: Tree drawings revisited. In: Speckmann, B., Tóth, C.D. (eds.) 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), vol. 99, pp. 23:1–23:15. Schloss Dagstuhl-Leibniz-Zentrum fuer Informatik, Dagstuhl (2018).
  6. 6.
    Crescenzi, P., Battista, G.D., Piperno, A.: A note on optimal area algorithms for upward drawings of binary trees. Comput. Geom. 2(4), 187–200 (1992). Scholar
  7. 7.
    Díaz, J., Petit, J., Serna, M.: A survey of graph layout problems. ACM Comput. Surv. 34(3), 313–356 (2002). Scholar
  8. 8.
    Dunham, W.: Euler : The Master of Us All. Dolciani Mathematical Expositions (Book 22). Mathematical Association of America, Washington (1999)zbMATHGoogle Scholar
  9. 9.
    Garg, A., Rusu, A.: Straight-line drawings of general trees with linear area and arbitrary aspect ratio. In: Kumar, V., Gavrilova, M.L., Tan, C.J.K., L’Ecuyer, P. (eds.) ICCSA 2003. LNCS, vol. 2669, pp. 876–885. Springer, Heidelberg (2003). Scholar
  10. 10.
    Garg, A., Rusu, A.: Straight-line drawings of binary trees with linear area and arbitrary aspect ratio. J. Graph Algorithms Appl. 8(2), 135–160 (2004). Scholar
  11. 11.
    Karp, R.M.: Reducibility among combinatorial problems. In: Complexity of Computer Computations: Proceedings of a symposium on the Complexity of Computer Computations, pp. 85–103 (1972). Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Hugo A. Akitaya
    • 1
  • Maarten Löffler
    • 2
  • Irene Parada
    • 3
    Email author
  1. 1.Tufts UniversityMedfordUSA
  2. 2.Utrecht UniversityUtrechtThe Netherlands
  3. 3.Graz University of TechnologyGrazAustria

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