The average size of ordered binary subgraphs

  • Pieter H. Hartel
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 344)


To analyse the demands made on the garbage collector in a graph reduction system, the change in size of an average graph is studied when an arbitrary edge is removed. In ordered binary trees the average number of deleted nodes as a result of cutting a single edge is equal to the average size of a subtree. Under the assumption that all trees with n nodes are equally likely to occur, the expected size of a subtree is found to be approximately √πn. The enumeration procedure can be applied to graphs by considering spanning trees in which the nodes that were shared in the graph are marked in the spanning tree. A correction to the calculation of the average is applied by ignoring subgraphs that have a marked root. Under the same assumption as above the average size of a subgraph is approximately √πn-2(m+1), where m represents the number of shared nodes and mn.

Key words

binary graphs Catalan statistics combinator graph reduction subgraphs 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    H. P. Barendregt, The lambda calculus, its syntax and semantics, North Holland, Amsterdam (1984).Google Scholar
  2. 2.
    P. J. Landin, “The mechanical evaluation of expressions,” Computer Journal 6(4) pp. 308–320 (Jan. 1964).Google Scholar
  3. 3.
    C. P. Wadsworth, Semantics and pragmatics of the lambda calculus, Oxford University, U.K. (1971). PhD. ThesisGoogle Scholar
  4. 4.
    P. H. Hartel, “A comparative study of garbage collection algorithms,” PRM project internal report D-23, Computing Science Department, University of Amsterdam (Feb. 1988).Google Scholar
  5. 5.
    J. Cohen, “Garbage collection of linked structures”, Computing Surveys 13(3) pp. 341–367 (Sep. 1981).Google Scholar
  6. 6.
    N. G. de Bruijn, D. E. Knuth, and S. O. Rice, “The average height of planted plane trees,” pp. 15–22 in Graph Theory and Computing, ed. R.C. Read, Academic Press, London, U.K. (1972).Google Scholar
  7. 7.
    P. H. Hartel and A. H. Veen, “Statistics on graph reduction of SASL programs,” Software practice and experience 18(3) pp. 239–253 (Mar. 1988).Google Scholar
  8. 8.
    D. E. Knuth, The art of computer programming, volume 1: Fundamental algorithms, Addison Wesley, Reading, Massachusetts (1973). second editionGoogle Scholar
  9. 9.
    H. W. Gould and J. Kaucky, “Evaluation of a class of binomial coefficient summations,” Journal of Combinatorial theory 1(2) pp.233–247 (Sep. 1966).Google Scholar
  10. 10.
    D. E. Knuth, The art of computer programming, volume 2: Seminumerical algorithms, Addison Wesley, Reading, Massachusetts (1980). second editionGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1989

Authors and Affiliations

  • Pieter H. Hartel
    • 1
  1. 1.Computing Science DepartmentUniversity of AmsterdamAmsterdam

Personalised recommendations