Summary: If \(X\) is a finite tree and \(f \colon X \rightarrow X\) is a map, in the Main Theorem of this paper (Theorem 1.8), we find eight conditions, each of which is equivalent to \(f\) being equicontinuous. To name just a few of the results obtained: the equicontinuity of \(f\) is equivalent to there being no arc \(A \subseteq X\) satisfying \(A \subsetneq f^n [A]\) for some \(n\in \mathbb{N}\). It is also equivalent to the statement that for some nonprincipal ultrafilter \(u\), the function \(f^u \colon X \rightarrow X\) is continuous (in other words, failure of equicontinuity of \(f\) is equivalent to the failure of continuity of every element of the Ellis remainder \(g\in E(X,f)^{\ast})\). One of the tools used in the proofs is the Ramsey-theoretic result known as Hindman’s theorem. Our results generalize the ones shown by I. Vidal-Escobar and S. Garcia-Ferreira [Topology Appl. 265, Article ID 106756, 8 p. (2019; Zbl 1426.37012)], and complement those of A. M. Bruckner and J. Ceder [Pac. J. Math. 156, No. 1, 63–96 (1992; Zbl 0728.58020)], J.-H. Mai [Trans. Am. Math. Soc. 355, No. 10, 4125–4136 (2003; Zbl 1026.54031)] and J. Camargo et al. [Chaos Solitons Fractals 126, 1–6 (2019; Zbl 1448.54023)].