Let \(G\) be a simple graph on \([n] = \{1, \ldots,n\}\) and \(S\) a polynomial ring in \(2n\) variables over a field. In this paper, the authors focus their attention on the regularity of the class of binomial edge ideals. Such a class of ideals, introduced independently by J. Herzog et al. [Adv. Appl. Math. 45, No. 3, 317–333 (2010; Zbl 1196.13018)] and M. Ohtani [Commun. Algebra 39, No. 3, 905–917 (2011; Zbl 1225.13028)], is “a generalization of the classical determinantal ideal generated by the 2-minors of a \(2\times n\) matrix of indeterminates”. The authors show that if \(G\) is a closed graph, then “the regularity of the binomial edge ideal \(J_G\) coincides with the regularity of the initial ideal of \(J_G\)”, with respect to the lexicographic order on \(S\) and “can be expressed in terms of the combinatorial data of \(G\)”. K. Matsuda and S. Murai [J. Commut. Algebra 5, No. 1, 141–149 (2013; Zbl 1272.13018)] conjectured that if \(G\) is a simple graph on \([n]\) such that the regularity of \(J_G\) equals \(n\), then \(G\) is a path of length \(n\). The authors of this paper give a positive answer to such a conjecture for a class of chordal graphs including trees, deriving, as a consequence, that the assertion holds for trees.