(Electronic Journal of Combinatorics, 2017) Freiberg, Uta; Heuberger, Clemens; Prodinger, Helmut

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Consider infinite random words over a finite alphabet where the letters occur
as an i.i.d. sequence according to some arbitrary distribution on the alphabet. The
expectation and the variance of the waiting time for the first completed h-run of
any letter (i.e., first occurrence of h subsequential equal letters) is computed.
The expected waiting time for the completion of h-runs of j arbitrary distinct
letters is also given.

(SIAM, 2015) Heuberger, Clemens; Krenn, Daniel; Wagner, Stephan

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For fixed t ≥ 2, we consider the class of representations of 1 as a sum of unit fractions
whose denominators are powers of t, or equivalently the class of canonical compact t-ary Huffman
codes, or equivalently rooted t-ary plane “canonical” trees. We study the probabilistic behavior of
the height (limit distribution is shown to be normal), the number of distinct summands (normal
distribution), the path length (normal distribution), the width (main term of the expectation and
concentration property), and the number of leaves at maximum distance from the root (discrete
distribution).

(University of Belgrade - School of Electrical Engineering, 2017-10) Heuberger, Clemens; Prodinger, Helmut

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The protection number of a plane tree is the minimal distance of the root
to a leaf; this definition carries over to an arbitrary node in a plane tree by
considering the maximal subtree having this node as a root. We study the the
protection number of a uniformly chosen random tree of size n and also the
protection number of a uniformly chosen node in a uniformly chosen random
tree of size n. The method is to apply singularity analysis to appropriate
generating functions. Additional results are provided as well.