Let $T$ be a tree with $n$ vertices. Let $f: T \rightarrow T$ be
continuous and suppose that the $n$ vertices form a periodic orbit
under $f$. The combinatorial information that comes from possible
permutations of the vertices gives rise to an irreducible
representation of $S_n$. Using the algebraic information it is
shown that $f$ must have periodic orbits of certain periods.
Finally, a family of maps is defined which shows that the result
about periods is best possible
if $n=2^k+2^l$ for $k, l \geq 0$.