Contraction Displays on Labeled Trees: Bounded Collision Cores, Lower Shadows, and Exponential Containment

Abstract: Let $T_m$ be the set of labeled trees on $[m]$, and for a fixed labeled tree $T\in T_n$ let [ μ_T(m)=#{U\in T_m:T\preceq U} ] be the number of labeled trees on $[m]$ that display $T$ as a contraction, with labels standardized after contraction. Motivated by a Reversal-Wilf problem for this support-count sequence, we study contraction displays through marked displays and collision moments. The main results are: a survivor split-system criterion for $T\preceq U$; a closed formula for the marked display count $C_1(T;m)$ in terms of the degree sequence of $T$; a bounded collision-core theorem showing that every $k$-overlay state of an $n$-vertex tree contracts to a core with at most $k(n-1)+1$ vertices; a contraction-diamond theorem showing that every lower one-edge collision is realized as the shadow of a bounded pair-core; and the exponential containment estimate [ μ_T(m)=m^{m-2}\left(1-O_T(e^{-c_Tm})\right) ] for every fixed labeled tree $T$ and some $c_T>0$. We also give exact formulas and an exponential generating function for a top-centered star-avoidance refinement. Together these results give a finite collision-core framework for the Reversal-Wilf problem. The remaining questions concern whether support-count equality determines the relevant collision profiles and how much lower contraction data can be decoded from the bounded core profiles.