Abstract
In this paper we generalize a result of de Bruijn, Knuth und Rice concerning the average height of planted plane trees withn nodes. First we compute the number of allr-typly rooted planted plane trees (r-trees) withn nodes and height less than or equal tok. Assuming that all planted plane trees withn nodes are equally likely, we show, that in the average a planted plane tree is a 3-tree for largen; for this distribution we compute also the cumulative distribution function and the variance. Finally, we shall derive an exact expression and its asymptotic equivalent for the average height\(\bar h_r \) (n) of anr-tree withn nodes. We obtain for all ε>0
Zusammenfassung
Wir verallgemeinern in dieser Arbeit ein Ergebnis von de Bruijn, Knuth und Rice über die Höhe planarer Wurzelbäume mitn Knoten. Wir berechnen zunächst die Anzahl allerr-fach gewurzelter planarer Bäume (r-Bäume) mitn Knoten und einer Höhe kleiner gleichk. Unter der Annahme, daß alle planare Bäume mitn Knoten gleichwahrscheinlich sind, zeigen wir, daß für großen ein planarer Wurzelbaum ein 3-Baum ist; für diese Verteilung berechnen wir die Verteilungsfunktion und die Varianz. Schließlich leiten wir einen exakten Ausdruck und sein asymptotisches Äquivalent für die mittlere Höhe\(\bar h_r \) n einesr-Baumes mitn Knoten ab. Wir erhalten für alle ε>0
Similar content being viewed by others
References
Abramowitz, M., Stegun, I. A.: Handbook of mathematical functions. New York: Dover Publications 1970.
Apostol, T. M.: Introduction to analytic number theory. New York: Springer 1976.
De Bruijn, N. G., Knuth, D. E., Rice, S. O.: The average height of planted plane trees, in: Graph theory and computing (R. C. Read, ed.), pp. 15–22. New York-London: Academic Press 1972.
Flajolet, Ph., Raoult, J. C., Vuillemin, J.: On the average number of registers required for evaluating arithmetic expressions. IRIA Rapport de Recherche, No. 228 (1977).
Kemp, R.: The average number of registers needed to evaluate a binary tree optimally. Acta Informatica11, 363–372 (1979).
Kemp, R.: On the average stack size of regularly distributed binary trees, in: Proc. of the sixth international colloquium on automata, languages and programming (ICALP 79), pp. 340–355. Berlin-Heidelberg-New York: Springer 1979.
Kemp, R.: The average depth of a prefix of the Dycklanguage D1, in: Proc. of the 2-th international conference of fundamentals of computing theory (FCT 79), pp. 230–236 (1979).
Knuth, D. E.: The art of computer programming, Vol. 1, 2nd ed., Reading, Mass.: Addison-Wesley 1973.
Kreweras, G.: Sur les éventails de segments. Cahiers du B.U.R.O.15, 1–41 (1970).
Kuich, W., Prodinger, H., Urbanek, F. J.: On the height of derivation trees, in: Proc. of the 6th international colloquium on automata, languages and programming (ICALP 79), pp. 370–384. Berlin-Heidelberg-New York: Springer 1979.
Munro, I.: Random walks in binary trees. CS-Dept., University of Waterloo, 1976.
Prodinger, H.: The average maximal lead position of a Ballot sequence. Preprint, Technische Universität Wien, 1979.
Riordan, J.: Combinatorial identities. New York: Wiley 1968.
Ruskey, F., Hu, T. C.: Generating binary trees lexicographically. Siam J. Comput.6, 745–758 (1977).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kemp, R. The average height of r-typly rooted planted plane trees. Computing 25, 209–232 (1980). https://doi.org/10.1007/BF02242000
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF02242000