  Given a tree map $f:T\to T$, we study the dynamics of subcontinua of $T$
under action of $f$. In particular, we prove that a subcontinuum of $T$ is
either asymptotically periodic or asymptotically degenerate. As an application
of this result, we show that zero topological entropy of the system $(T,f)$
implies zero topological entropy of its functional envelope (endowed with the
Hausdorff metric).
