Abstract
For fixed $t\ge 2$, we consider the class of representations of $1$ as 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 behaviour 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).
Highlights
We consider three combinatorial classes, which all turn out to be equivalent: partitions of 1 into powers of t, canonical compact t-ary Huffman codes, and “canonical” t-ary trees; see the precise discussion below
We are interested in the structure of these objects under a uniform random model, and we study the distribution of various structural parameters, for which we obtain rather precise limit theorems
We study the number of distinct depths of leaves d(T ) of a canonical tree T ∈ T, motivated by the interpretation as the number of distinct code word lengths in Huffman codes. This parameter is asymptotically normally distributed, and we show a local limit theorem
Summary
We consider three combinatorial classes, which all turn out to be equivalent: partitions of 1 into powers of t, canonical compact t-ary Huffman codes, and “canonical” t-ary trees; see the precise discussion below. We will be able to apply singularity analysis to all our generating functions in the coming sections At this point, we restate Theorem 1.1 on the number of trees taking the notation of Theorem 2.1 into account and extend it to the number of canonical forests with r roots. By singularity analysis [13, 15], Lemma 2.4, and Theorem 2.1, the number of canonical forests with r roots of size n is (2.14). When analyzing the asymptotic behavior of the height (section 3), the number of leaves on the last level (section 6), and the path length (section 7), the corresponding formulae contain the infinite sum b(q, u, 1, w) and its derivatives.
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