Constructing and searching conditioned Galton-Watson trees

TL;DR

This paper studies how to construct and analyze conditioned Galton-Watson trees, revealing their structure and offspring distributions, and applies these findings to compute the expected search cost in such trees.

Contribution

It explicitly characterizes the structure of conditioned Galton-Watson trees and derives formulas for their offspring distributions and search costs.

Findings

Conditioned Galton-Watson trees have level-dependent offspring distributions.

Explicit formulas for offspring distributions of conditioned trees.

Calculated expected search cost in conditioned Galton-Watson trees.

Abstract

We investigate conditioning Galton-Watson trees on general recursive-type events, such as the event that the tree survives until a specific level. It turns out that the conditioned tree is again a type of Galton-Watson tree, with different types of offspring and a level-dependent offspring distribution, which will all be given explicitly. As an interesting application of these results, we will calculate the expected cost of searching a tree until reaching a given level.