This paper discuss a problem of locating a facility on tree networks. The facility takes the form of a path- or a tree-shaped subgraph on the network. To construct efficient network, we must set the facility to optimal location. In this paper, we propose new evaluation function, all-pair distancesum, which is the sum of distance of all pairs of nodes on the network. Then we show some characteristics of it, which lead to the following relations. In case of continuous tree-shaped version, minimum all-pair distancesum facility is same to minimum distancesum facility. In case of path-shaped or descrete tree-shaped versions, these facilities may be different. Furthermore, we show how to locate minimum all-pair distancesum facility.