An infinite homogeneous tree is a special type of graph that has a completely symmetrical structure in all directions. For an infinite homogeneous tree T=(V,E) with the natural distance d defined on graphs and a weighted measure μ of exponential growth, the authors introduce the variable Lebesgue space Lp(·)(μ) over (V,d,μ) and investigate it under the global Hölder continuity condition for p(·). As an application, the strong and weak boundedness of the maximal operator relevant to admissible trapezoids on Lp(·)(μ) is obtained, and an unbounded example is presented.