A note on the size of query trees

TL;DR

This paper provides simple proofs showing that query trees in graphs with bounded degree have expected constant size and logarithmic size with high probability, depending on the degree bound and number of nodes.

Contribution

It offers straightforward proofs establishing bounds on query tree sizes in bounded degree graphs, clarifying their expected and high-probability sizes.

Findings

Expected size of query trees is constant.

Query tree size is logarithmic with high probability.

Size bounds depend on the degree bound and number of nodes.

Abstract

We consider query trees of graphs with degree bounded by a constant, $d$. We give simple proofs that the size of a query tree is constant in expectation and $2^{O(d)}\log{n}$ w.h.p.