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# The expected number of nodes and leaves at level k in ordered trees

Contributed Papers
Part of the Lecture Notes in Computer Science book series (LNCS, volume 145)

## Abstract

In this paper the average number of nodes (nodes of degree t) appearing at some level k in a n-node ordered tree is computed. Exact enumeration formulae for the number of all n-node trees with r nodes (r nodes of degree t) at level k are derived. Assuming all n-node trees to be equally likely, it is shown that the expected number n1(n,k) of nodes (n 1 (t) (n,k) of nodes of degree t) at level k is given by
$$\begin{gathered}n_1 (n,k) = {{n\tfrac{{2k - 1}}{{2n - 1}}\left( {\begin{array}{*{20}c}{2n - 1} \\{n - k} \\\end{array} } \right)} \mathord{\left/{\vphantom {{n\tfrac{{2k - 1}}{{2n - 1}}\left( {\begin{array}{*{20}c}{2n - 1} \\{n - k} \\\end{array} } \right)} {\left( {\begin{array}{*{20}c}{2n - 2} \\{n - 1} \\\end{array} } \right)}}} \right.\kern-\nulldelimiterspace} {\left( {\begin{array}{*{20}c}{2n - 2} \\{n - 1} \\\end{array} } \right)}} \hfill \\and \hfill \\n_{_1 }^{(t)} (n,k) = {{n\tfrac{{2k + t - 1}}{{2n - t - 1}}\left( {\begin{array}{*{20}c}{2n - t - 1} \\{n - t - k} \\\end{array} } \right)} \mathord{\left/{\vphantom {{n\tfrac{{2k + t - 1}}{{2n - t - 1}}\left( {\begin{array}{*{20}c}{2n - t - 1} \\{n - t - k} \\\end{array} } \right)} {\left( {\begin{array}{*{20}c}{2n - 2} \\{n - 1} \\\end{array} } \right)}}} \right.\kern-\nulldelimiterspace} {\left( {\begin{array}{*{20}c}{2n - 2} \\{n - 1} \\\end{array} } \right)}}. \hfill \\\end{gathered}$$

Furthermore, asymptotic equivalents for these expected values and exact expressions for the higher moments about the origin are computed.

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## References

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## Copyright information

© Springer-Verlag Berlin Heidelberg 1982

## Authors and Affiliations

• R. Kemp
• 1
1. 1.Fachbereich Informatik (20)Johann Wolfgang Goethe-UniversitätFrankfurt a. M.