The authors show that the coefficient matrix of a quadratic \(M\)-convex function can be expressed by the distance matrix of some tree metric and emphasize the tree representation of a quadratic \(M\)-convex function. \textit{A. Dress}, \textit{V. Moulton} and \textit{W. Terhalle} [Eur. J. Comb. 17, No. 2--3, 161--175 (1996; Zbl 0853.54027)] have made some interesting observations regarding the valuated matroids of rank two and indicated their close relationship to the four-point condition of Buneman characterizing a tree metric. The authors claim that their result may be regarded as an extension of this observation of Dress et al. (loc. cit.) to \(M\)-convex functions. It is also shown that the discrete Hessian matrix of an \(M\)-convex function is expressed by the distance matrix of some tree metric and consequently an \(M\)-convex function can be expanded locally to quadratic \(M\)-convex function.