Any tree, with \(n\) edges and \(n + 1\) vertices, can be realized in Euclidean \(n\)-space so that its edges, of any chosen lengths, are mutually perpendicular. The convex hull of such an orthogonal tree is an orthogonal simplex whose dihedral angles include \((\begin{smallmatrix} n \\2 \end{smallmatrix})\) right angles. More precisely, each vertex of the tree, being also a vertex of the simplex, represents (as in a Coxeter-Dynkin diagram) the opposite facet of the simplex. The two ends of an edge of the tree represent two facets forming an acute dihedral angle; each of the remaining \((\begin{smallmatrix} n \\2 \end{smallmatrix})\) pairs of facets are orthogonal. This happens because, for any two nonadjacent vertices of the tree, the minimal subgraph joining them determines a simplex whose first and last facets are orthogonal. Since the remaining edges of the tree are orthogonal to the subspace spanned by the orthoscheme, these ``first and last facets'' are sections of orthogonal facets of the whole \(n\)-simplex.