Transversals in trees

TL;DR

This paper introduces a partial order on rooted trees based on the number of transversals of different sizes and characterizes the minimal element among trees with bounded degree.

Contribution

It defines a new partial order on rooted trees based on transversal counts and identifies the unique minimal tree within degree constraints.

Findings

Existence of a unique minimal tree for each degree and size

Characterization of the minimal tree structure

Comparison of trees based on transversal counts

Abstract

A transversal in a rooted tree is any set of nodes that meets every path from the root to a leaf. We let c(T,k) denote the number of transversals of size k in a rooted tree T. We define a partial order on the set of all rooted trees with n nodes by saying that a tree T succeeds a tree T' if c(T,k) is at least c(T',k) for all k and strictly greater than c(T',k) for at least one k. We prove that, for every choice of positive integers d and n, the set of all rooted trees on n nodes where each node has at most d children has a unique minimal element with respect to this partial order and we describe this tree.