Trees With Path-Stable Center

Abstract

We study the notion of path-congruence $\mathrm{\Phi }:T_1\to T_2$ between two trees $T_1$ and $T_2$. We introduce the concept of the trunk of a tree, and prove that, for any tree $T$, the trunk and the periphery of $T$ are stable. We then give conditions for which the center of $T$ is stable. One such condition is that the central vertices have degree $2$. Also, the center is stable when the diameter of $T$ is less than $8$.