Efficient parallel algorithms for finding an optimal path-shaped or tree-shaped facility with a specified size in a tree network are presented. Four kinds of optimization criteria are considered: minimizing/maximizing distancesum/eccentricity. There are eight cases when considering facility shapes and optimization criteria. Parallel algorithms for finding a minimum/maximum distancesum path were presented in [9]. The other six cases are studied in this paper. Two of these six cases can be solved optimally in linear Time x Processor complexity. For the problem of finding a maximum distancesum tree, the algorithm presented in this paper is the first polynomial time solution under a reasonable assumption.