Open and increasing paths on N-ary trees with different fitness values

TL;DR

This paper studies the probabilistic behavior of open and increasing paths in N-ary trees with random fitness values, analyzing their counts and lengths under various stochastic models.

Contribution

It introduces new results on the limiting behavior of open and increasing paths in N-ary trees with random variables, including asymptotic properties.

Findings

Distribution of the number of open paths from root to leaves

Asymptotic behavior of the longest open path

Properties of the longest increasing path with continuous random variables

Abstract

Consider a rooted N-ary tree. For every vertex of this tree, we atttach an i.i.d. Bernoulli random variable. A path is called open if all the random variables that are assigned on the path are 1. We consider limiting behaviors for the number of open paths from the root to leaves and the longest open path. In addition, when all fitness values are i.i.d. continuous random variables, some asymptotic properties of the longest increasing path are proved.