By a tree we mean a connected space that is a finite union of closed intervals and contains no circle. The simplest example to keep in mind is a closed interval itself. Given a continuous map f from a tree into itself, we investigate how closed connected subsets of the tree behave under the action of f. We prove that every such a set, when iterated under f, is either asymtotically periodic or asymtotically degenerate. As an important corollary, we prove that whenever f has zero entropy, so does the functional envelope of f. By the functional envelope of f we mean the dynamical system on the set of all maps from the tree into itself, where each map g is sent to the composition f(g).