This paper studies the integrated problem of single machine scheduling and location on a tree graph, where the processing times of jobs, transportation starting times, and the transportation speeds of jobs are given; the vertices and edges of the tree graph are also given, but the edge lengths are uncertain and their values depend on a finite set of discrete scenarios. The placement of the machine and the processing sequence of jobs need to be determined by decision-making; jobs need to be transported to the machine location before processing. The scheduling criterion of interest is the maximum completion time (or makespan) of jobs. Using robust optimization methods, polynomial-time algorithms are provided for the maximum absolute robust problem and the maximum regret problem respectively; for the maximum relative regret problem, a fully polynomial-time approximation scheme is designed. Finally, numericals examples are given to illustrate the execution process and effectiveness of the proposed algorithms.