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The average height of r-typly rooted planted plane trees

Die mittlere Höhe r-fach gewurzelter planarer Bäume

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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

$$\bar h_r (n) = \sqrt {\pi n} - \frac{1}{2}(r - 2) + O(1n(n)/n^{1/2 - \varepsilon } ).$$

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

$$\bar h_r (n) = \sqrt {\pi n} - \frac{1}{2}(r - 2) + O(1n(n)/n^{1/2 - \varepsilon } ).$$

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Kemp, R. The average height of r-typly rooted planted plane trees. Computing 25, 209–232 (1980). https://doi.org/10.1007/BF02242000

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Keywords

  • Distribution Function
  • Plane Tree
  • Computational Mathematic
  • Cumulative Distribution
  • Cumulative Distribution Function