Contraction Containment in Labeled Trees: Support Counts, Collision Cores, and Star Avoidance

Abstract

We study whether a labeled tree can be recovered from the numbers of larger labeled trees that contract to it. We first characterize containment using the edge splits induced on the surviving vertices after contraction and derive an exact formula for containing trees with one marked display. We then study several displays in the same containing tree simultaneously. After contracting every edge invisible to all \(k\) displays of an \(n\)-vertex target, the remaining collision core has at most \(k(n-1)+1\) vertices. Once the coincidences and relative positions of the surviving labels are recorded, these bounded cores give a finite exact expansion of each collision count. As applications, we prove that almost every sufficiently large labeled tree contains any fixed target, with an exponentially small exceptional proportion. We also obtain an exact avoidance formula and generating function for stars whose center is represented by the largest host label, together with fixed-star asymptotics and a coarse threshold.