A Brief History of Strahler Numbers

  • Javier Esparza
  • Michael Luttenberger
  • Maximilian Schlund
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8370)


The Strahler number or Horton-Strahler number of a tree, originally introduced in geophysics, has a surprisingly rich theory. We sketch some milestones in its history, and its connection to arithmetic expressions, graph traversing, decision problems for context-free languages, Parikh’s theorem, and Newton’s procedure for approximating zeros of differentiable functions.


Binary Tree Newton Iteration Derivation Tree Arithmetic Expression Search Number 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Bienstock, D., Robertson, N., Seymour, P., Thomas, R.: Quickly excluding a forest. Journal of Combinatorial Theory, Series B 52(2), 274–283 (1991), CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Bozapalidis, S.: Equational elements in additive algebras. Theory Comput. Syst. 32(1), 1–33 (1999)CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Brázdil, T., Esparza, J., Kiefer, S., Luttenberger, M.: Space-efficient scheduling of stochastically generated tasks. Inf. Comput. 210, 87–110 (2012)CrossRefzbMATHGoogle Scholar
  4. 4.
    Chytil, M., Monien, B.: Caterpillars and context-free languages. In: Choffrut, C., Lengauer, T. (eds.) STACS 1990. LNCS, vol. 415, pp. 70–81. Springer, Heidelberg (1990)CrossRefGoogle Scholar
  5. 5.
    Devroye, L., Kruszewski, P.: A note on the Horton-Strahler number for random trees. Inf. Process. Lett. 56(2), 95–99 (1995)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Droste, M., Kuich, W., Vogler, H.: Handbook of Weighted Automata. Springer (2009)Google Scholar
  7. 7.
    Ehrenfeucht, A., Rozenberg, G., Vermeir, D.: On et0l systems with finite tree-rank. SIAM J. Comput. 10(1), 40–58 (1981)CrossRefzbMATHMathSciNetGoogle Scholar
  8. 8.
    Ershov, A.P.: On programming of arithmetic operations. Comm. ACM 1(8), 3–9 (1958)CrossRefzbMATHGoogle Scholar
  9. 9.
    Esparza, J., Kiefer, S., Luttenberger, M.: On fixed point equations over commutative semirings. In: Thomas, W., Weil, P. (eds.) STACS 2007. LNCS, vol. 4393, pp. 296–307. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  10. 10.
    Esparza, J., Kiefer, S., Luttenberger, M.: Computing the least fixed point of positive polynomial systems. SIAM J. Comput. 39(6), 2282–2335 (2010)CrossRefzbMATHMathSciNetGoogle Scholar
  11. 11.
    Esparza, J., Kiefer, S., Luttenberger, M.: Newtonian program analysis. J. ACM 57(6), 33 (2010)CrossRefMathSciNetGoogle Scholar
  12. 12.
    Esparza, J., Ganty, P., Kiefer, S., Luttenberger, M.: Parikhs theorem: A simple and direct automaton construction. Inf. Process. Lett. 111(12), 614–619 (2011)CrossRefzbMATHMathSciNetGoogle Scholar
  13. 13.
    Esparza, J., Ganty, P., Majumdar, R.: Parameterized verification of asynchronous shared-memory systems. In: Sharygina, Veith (eds.) [24], pp. 124–140Google Scholar
  14. 14.
    Esparza, J., Luttenberger, M.: Solving fixed-point equations by derivation tree analysis. In: Corradini, A., Klin, B., Cîrstea, C. (eds.) CALCO 2011. LNCS, vol. 6859, pp. 19–35. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  15. 15.
    Etessami, K., Yannakakis, M.: Recursive markov chains, stochastic grammars, and monotone systems of nonlinear equations. J. ACM 56(1) (2009)Google Scholar
  16. 16.
    Flajolet, P., Raoult, J.-C., Vuillemin, J.: The number of registers required for evaluating arithmetic expressions. Theor. Comput. Sci. 9, 99–125 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Flajolet, P., Prodinger, H.: Register allocation for unary-binary trees. SIAM J. Comput. 15(3), 629–640 (1986)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Ganty, P., Majumdar, R., Monmege, B.: Bounded underapproximations. Formal Methods in System Design 40(2), 206–231 (2012)CrossRefzbMATHGoogle Scholar
  19. 19.
    Ginsburg, S., Spanier, E.: Derivation-bounded languages. Journal of Computer and System Sciences 2, 228–250 (1968)CrossRefzbMATHMathSciNetGoogle Scholar
  20. 20.
    Horton, R.E.: Erosional development of streams and their drainage basins: hydro-physical approach to quantitative morphology. Geol. Soc. Am. Bull. 56(3), 275–370 (1945)CrossRefGoogle Scholar
  21. 21.
    Kemp, R.: The average number of registers needed to evaluate a binary tree optimally. Acta Informatica 11, 363–372 (1979)CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Megiddo, N., Hakimi, S.L., Garey, M.R., Johnson, D.S., Papadimitriou, C.H.: The complexity of searching a graph (preliminary version). In: FOCS, pp. 376–385. IEEE Computer Society (1981)Google Scholar
  23. 23.
    Pivoteau, C., Salvy, B., Soria, M.: Algorithms for combinatorial structures: Well-founded systems and newton iterations. J. Comb. Theory, Ser. A 119(8), 1711–1773 (2012)CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Sharygina, N., Veith, H. (eds.): CAV 2013. LNCS, vol. 8044. Springer, Heidelberg (2013)zbMATHGoogle Scholar
  25. 25.
    Stewart, A., Etessami, K., Yannakakis, M.: Upper Bounds for Newton’s Method on Monotone Polynomial Systems, and P-Time Model Checking of Probabilistic One-Counter Automata. In: Sharygina, Veith (eds.) [24], pp. 495–510Google Scholar
  26. 26.
    Strahler, A.N.: Hypsometric (area-altitude) analysis of erosional topology. Geol. Soc. Am. Bull. 63(11), 1117–1142 (1952)CrossRefGoogle Scholar
  27. 27.
    Yntema, M.K.: Inclusion relations among families of context-free languages. Information and Control 10, 572–597 (1967)CrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Javier Esparza
    • 1
  • Michael Luttenberger
    • 1
  • Maximilian Schlund
    • 1
  1. 1.Fakultät für InformatikTechnische Universität MünchenGermany

Personalised recommendations