The purpose of this paper is to prove the following theorem: If a semitopological semigroup \(S\) is defined on a finite tree and if \(S^2=S\), then \(S\) is a topological semigroup. Let \(M(S)\) denote the minimal ideal of \(S\), and \(m(s)\) the single element contained in the intersection of all arcs \([m,s]\cap M(S)\). To prove the theorem, the following results are used. Let \(S\) be a semitopological semigroup defined on a tree and \(e,f\) be idempotent elements of \(S\). (1) If the minimal ideal of \(S\) consists of left zeros, then \(x[m(e),e]= [xm(e),x]\) for every \(x\in fSe\). (2) The arc \([m(e),e]\) is an \(I\)-semigroup with identity element \(e\) and zero \(m(e)\). (3) The set \(S(e)=M(S) \cup[m(e),e]\) is a topological semigroup. The theorem is a generalization of \textit{J. F. Berglund}'s result [J. Lond. Math. Soc., II. Ser. 4, 533-540 (1972; Zbl 0238.22003)] with respect to an interval \(S\) satisfying \(S^2=S\).