Let $X$ be a compact tree, $f$ be a continuous map from $X$ to itself,
$End(X)$ be the number of endpoints and $Edg(X)$ be the number of edges of $X$.
We show that if $n>1$ has no prime divisors less than $End(X)+1$ and $f$ has a
cycle of period $n$, then $f$ has cycles of all periods greater than
$2End(X)(n-1)$ and topological entropy $h(f)>0$; so if $p$ is the least prime
number greater than $End(X)$ and $f$ has cycles of all periods from 1 to
$2End(X)(p-1)$, then $f$ has cycles of all periods (this verifies a conjecture
of Misiurewicz for tree maps). Together with the spectral decomposition theorem
for graph maps it implies that $h(f)>0$ iff there exists $n$ such that $f$ has
a cycle of period $mn$ for any $m$. We also define {\it snowflakes} for tree
maps and show that $h(f)=0$ iff every cycle of $f$ is a snowflake or iff the
period of every cycle of $f$ is of form $2^lm$ where $m\le Edg(X)$ is an odd
integer with prime divisors less than $End(X)+1$.
