Structure, Enumeration, and star avoidance for contraction containment in labeled trees

Abstract

We study contraction containment among labeled trees, where a labeled tree $U$ displays a fixed tree $T$ if $T$ is obtained by contracting connected fibers and standardizing the surviving labels. We develop a collision-core framework for the support counts $μ_T(m)$, prove a survivor split-system criterion and a degree-sequence formula for the marked display count $C_1(T;m)$, and show that every $k$-overlay of an $n$-vertex tree contracts to a reduced core with at most $k(n-1)+1$ vertices. For each reduced core we give an exact lift formula and product exponential generating function for all overlays lying above it, yielding finite formulas for fixed-order collision moments. We also prove a contraction-diamond theorem showing that every lower one-edge collision is realized as the shadow of a bounded pair-core. In a complementary exact-enumeration direction, we solve top-centered star avoidance: if $S_n$ is the star on $[n+1]$ centered at $n+1$, then \[ A_n^{\mathrm{top}}(k)=\sum_{L=0}^{n-1}τ(k,L)+\frac{n}{k}τ(k,n), \] where $τ(k,L)$ counts trees on $[k]$ with $L$ leaves. This gives rational-logarithmic generating functions, sharp fixed-$n$ asymptotics, and a random-tree threshold at $n\sim k/e$. Finally, every fixed labeled tree is displayed by almost every large labeled tree: \[ μ_T(m)=m^{m-2}\bigl(1-O_T(e^{-c_Tm})\bigr). \]