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 2lm where m≤Edg(X) is an odd
integer with prime divisors less than End(X)+1