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

TL;DR

This paper establishes mathematical relations between the properties of complete m-ary rooted trees and the distribution of lengths of strings generated by a specific stochastic process, linking tree metrics to string length expectations and variances.

Contribution

It proves new formulas connecting tree structure metrics with the expectation and variance of string lengths generated by a Markov process, providing novel interpretations of known integer sequences.

Findings

Expected string length equals total number of edges in the tree

Variance of string length relates to the sum of common path lengths in the tree

New interpretation of OEIS sequence A286778 as variance of coin tosses until n heads in a row

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.