# Dependence between Path-length and Size in Random Digital Trees

## TL;DR

This paper investigates the dependence between size and path length in random digital trees, revealing asymptotic independence in asymmetric cases and strong dependence with fluctuations in symmetric cases, contrasting prior results.

## Contribution

It uncovers novel dependence behaviors in digital trees, showing independence or dependence based on symmetry, and extends findings to other digital tree classes.

## Key findings

- Asymptotic independence in asymmetric digital tries.
- Strong dependence with periodic fluctuations in symmetric tries.
- Different correlation behaviors in various digital tree classes.

## Abstract

We study the size and the external path length of random tries and show that they are asymptotically independent in the asymmetric case but strongly dependent with small periodic fluctuations in the symmetric case. Such an unexpected behavior is in sharp contrast to the previously known results on random tries that the size is totally positively correlated to the internal path length and that both tend to the same normal limit law. These two dependence examples provide concrete instances of bivariate normal distributions (as limit laws) whose correlation is $0$, $1$ and periodically oscillating. Moreover, the same type of behaviors is also clarified for other classes of digital trees such as bucket digital trees and Patricia tries.