Functional equations for data structures

  • François Bergeron
  • Gilbert Labelle
  • Pierre Leroux
Contributed Papers Algorithms
Part of the Lecture Notes in Computer Science book series (LNCS, volume 294)


We show how tree-like data structures (B-trees, AVL trees, binary trees, etc. ...) can be characterized by functional equations in the context of the theory of species of structures which has been introduced as a conceptual framework for enumerative combinatorics. The generating functions associated to these abstract data structures are directly derived from the corresponding functional equations.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    M. Bajraktarevic: Sur une équation fonctionnelle, Glasnik Mat.-Fiz. I Astr. 12, 1957, pp 201–205.Google Scholar
  2. [2]
    R.P. Brent and H.T. Kung; Fast Algorithms for Manipulating Formal Power Series, JACM 25 (4), 1978, pp 581–595.Google Scholar
  3. [3]
    H. Décoste, G. Labelle, and P. Leroux; Une approche combinatoire pour l'itération de Newton-Raphson, Adv. in Appl. Math. 3, Acad. Press, 1982, pp 407–416.Google Scholar
  4. [4]
    S. Dubuc; Une équation fonctionnelle pour diverses constructions géométriques, Annales des Sciences mathématiques du Québec, Vol. 9, No.2, 1985, pp 151–174.Google Scholar
  5. [5]
    P. Flajolet; Elements of a General Theory of Combinatorial Structures, Proceedings FCT85,L. Budach, Ed., Lect. Notes in Comp. Science, Springer-Verlag, Vol. 199 (1985), pp. 112–127.Google Scholar
  6. [6]
    J. Françon: On the Analysis of Algorithms for Trees, Theor. Comp. Science, Vol 4, 1977, pp 155–169.Google Scholar
  7. [7]
    J. Françon; Sur le nombre de registres nécessaires à l'évaluation d'une expression arithmétique, RAIRO, Informatique théorique, 18 (1984), pp. 355–364.Google Scholar
  8. [8]
    G.H. Gonnet; Handbook of Algorithms and Data Structures, International Computer Science Series, Addison-Wesley, 1984.Google Scholar
  9. [9]
    A. Joyal; Une théorie combinatoire des séries formelles, Adv. in Math. 42, 1981, pp 1–82.Google Scholar
  10. [10]
    D. Knuth; The Art of Computer Programming, vol.2, Addison-Wesley, 1981.Google Scholar
  11. [11]
    G. Labelle; Some New Computational Methods in the Theory of Species, in Combinatoire énumérative, Proceedings, Montréal, Québec 1985, ed. G.Labelle and P.Leroux, Springer Lecture Notes in Math., No.1234, 1986, pp. 192–209.Google Scholar
  12. [12]
    G. Labelle; Une combinatoire sous-jacente au théorème des fonctions implicites, J. Compin. Theory, Series A, 40, 1985, pp 377–393Google Scholar
  13. [13]
    P. Leroux and G. Viennot; Combinatorial Resolution of Systems of Differential Equations I, Ordinary Differential Equations, in Combinatoire énumérative, Proceedings, Montréal, Québec 1985, ed. G.Labelle and P.Leroux, Springer Lecture Notes in Math., No.1234, 1986.Google Scholar
  14. [14]
    K. Mehlhorn; Data Structures and Algorithms, I: Sorting and Searching, Springer-Verlag, 1985.Google Scholar
  15. [15]
    T. Ottmann and D. Wood; 1-2 Brother Trees or AVL Trees Revisited, Comp. Jour., Vol 23 (3), 1980, pp 248–255.Google Scholar
  16. [16]
    A. Odlyzcko; Periodic Oscillations of coefficients of Power Series that satisfiy Functional equations, Adv. in Math., 44 (1982),pp. 180–205.Google Scholar
  17. [17]
    A.H. Read. The Solution of a Functional Equation, Proc. Royal Soc. Edinburgh, A 63, 1951–1952, pp 336–345.Google Scholar
  18. [18]
    J. Vuillemin; A Unifying Look at Data Structures, Comm. of ACM, Vol 23 (4), 1980, pp 229–239.Google Scholar
  19. [19]
    A.C-C. Yao; On Random 2–3 Trees, Acta Informatica, Vol. 9, 1978, pp 159–170.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1988

Authors and Affiliations

  • François Bergeron
    • 1
  • Gilbert Labelle
    • 1
  • Pierre Leroux
    • 1
  1. 1.Dép. de Mathématiques et d'InformatiqueUniversité du Québec à MontréalSucc. A, MontréalCanada

Personalised recommendations