Summary
We consider a general additive weight of random trees which depends on the structure of the subtrees, on weight functions defined on the number of internal and external nodes and on the degrees of the nodes appearing in the tree and its subtrees. Choosing particular weight functions, the corresponding weight is an important parameter appearing in the analysis of sorting and searching algorithms. For a simply generated family of rooted planar trees ℱ, we shall derive a general approach to the computation of the average weight of a tree Tεℱ with n nodes and m leaves for arbitrary weight functions. This general result implies exact and asymptotic expressions for many types of average weights of a tree Tεℱ with n nodes if the weight functions are arbitrary polynomials in the number of nodes and leaves with coefficients depending on the node degrees.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramowitz, M., Stegun, I.A.: Handbook, of Mathematical Functions. New York: Dover Publications 1970
Bender, E.A.: Asymptotic Methods in Enumeration. SIAM Rev. 16, 485–515 (1974)
Comtet, L.: Advanced Combinatorics. Dordrecht-Holland/Boston-USA: D. Reidel 1974
Dershowitz, N., Zaks, S.: Enumerations of Ordered Trees. Disc. Math. 31, 9–28 (1980)
Dershowitz, N., Zaks, S.: Applied Tree Enumeration. Lect. Notes Comput. Sci. 112 (6th CAAP), 180–193 (1981)
Flajolet, P., Odlyzko, A.: The Average Height of Binary Trees and Other Simple Trees. J. Comput. Sci. 25, 171–213 (1982)
Flajolet, P., Ottman, T., Wood, D.: Search Trees and Bubble Memories. R.A.I.R.O. Inf. Thèor./ Theor. Inf. 19, 137–164 (1985)
Kemp, R.: On The Average Oscillation of a Stack. Combinatorica 2, 157–176 (1982)
Kemp, R.: The Average Height of Planted Plane Trees with M. Leaves. J. Comb. Theory, Ser. B34, 191–208 (1983)
Kemp, R.: On a General Weight of Trees. Lect. Notes Comput. Sci. 166 (STACS 84), 109–120 (1984)
Kemp, R.: Fundamentals of The Average Case Analysis of Particular Algorithms. Wiley-Teubner Series in Computer Science. Stuttgart: Teubner, Chichester: Wiley 1984
Kemp, R.: Free Cost Measures of Trees. Lect. Notes Comput. Sci. 199 (FCT 85), 175–190 (1985)
Kemp, R.: The Analysis of an Additive Weight of Random Trees. Tech. Rep. 2/86, Universität Frankfurt, Fachbereich Informatik (1986)
Kemp, R.: On Systems of Additive Weights of Trees. Tech. Rep. 6/87, Universität Frankfurt, Fachbereich Informatik (1987)
Kemp, R.: Further Results on Leftist Trees. Tech. Rep. 7/87, Universität Frankfurt, Fachbereich Informatik (1987)
Kemp, R.: Additive Weights of Non-Regularly Distributed Trees. Ann. Discrete Math. 33, 129–155 (1987)
Knuth, D.E.: The Art of Computer Programming, Vol. 1, (2nd. ed.). Reading, Mass.: Addison Wesley 1973
Knuth, D.E.: The Art of Computer Programming, Vol. 3. Reading, Mass.: Addison Wesley 1973
Mahmoud, H.M.: On The Average Internal Path Length of M-ary Search Trees. Acta Inf. 23, 111–117 (1986)
Meir, M., Moon, J.W.: On The Altitude of Nodes in Random Trees. Can. Math. 30, 997–1015 (1973)
Roth, P.: Scharfe obere Schranken für freie Weglängen in Wurzelbäumen. Diplomarbeit, Universität Frankfurt, Fachbereich Informatik (1987)
Trier, U.: Additive Gewichte bei s-nären Leftist-Bäumen. Diplomarbeit, Universität Frankfurt, Fachbereich Informatik (1987)
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Kemp, R. The expected additive weight of trees. Acta Informatica 26, 711–740 (1989). https://doi.org/10.1007/BF00289158
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00289158