Let f: T → T be a tree map with n end-points, SAP(f) the set of strongly almost periodic points of f and CR(f) the set of chain recurrent points of f. Write E(f,T) = {x: there exists a sequence {ki} with 2 ≤ ki ≤ n such that and g = f\CR(f). In this paper, we show that the following three statements are equivalent:
(1) f has zero topological entropy.
(2) SAP(f) ⊂ E(f,T).
(3) Map ωg: x → ω(x,g) is continuous at p for every periodic point p of f.