Relations between the properties of a complete rooted tree and the properties of a distribution of lengths of randomly generated strings

Abstract

Let's denote a complete $m$-ary rooted tree graph of height $n$ as $G$. In scope of this paper we prove the certain relations between the properties of $G$ and the expectation and variance of the distribution of lengths of strings, generated as follows: starting from an empty string we pick a random symbol from the alphabet $\{ \alpha_1, \alpha_2, \dots \alpha_m \}$ and append it to the string, the process continues until we see $n$ instances of a specific symbol in a row. Consider a random variable $\xi_{m,n}$ that represents a length of a string generated according to the described process. The expectation $\mathbb{E}[\xi_{m,n}]$ and variance $\mathrm{Var}[\xi_{m,n}]$ depend on $m$ (the size of the alphabet) and $n$ (a parameter that defines a stopping criteria of the string generation process). Also, let's denote the sum of the common path length over all 2-tuples of nodes of $G$ as $S_{m,n}$, and let's denote the total number of edges in $G$ as $T_{m,n}$. In scope of this paper we prove that the following relations are true for all $m,n \geq 1$: $\mathbb{E}[\xi_{m,n}] = T_{m,n}$ and $\mathrm{Var}[\xi_{m,n}] = (m-1) \cdot S_{m,n}$. While it is known that both $\mathbb{E}[\xi_{2,n}]$ and $T_{2,n}$ are described by the sequence A000918 from the On-Line Encyclopedia of Integer Sequences (OEIS), and it is known that $S_{2,n}$ is described by the OEIS sequence A286778, we demonstrate a new interpretation for A286778: this sequence describes $\mathrm{Var}[\xi_{2,n}]$ - a variance of the number of tosses of a fair coin until we see $n$ heads in a row.