This paper addresses the robust balance point problem on a tree with interval data of vertex weights and edge lengths, the vanishing balance objective function on trees is defined to avoid the hardness of computing the classical balance value. We first show that the robust balance point is contained in an absolute subtree. Then we search for the optimal solution on the underlying subtree in two phases. The first phase is to compute the robust values at all vertices of the subtree. The second phase is to compute these values at five candidate points on the interior of each edge. As each of the two phases can be solved in linear time, the regarding robust balance point can be found with the similar complexity.