The uncertainty management problem for moving objects databases has been well studied recently, with many models and algorithms proposed. However, very limited work has dealt with the index of uncertain trajectories for a running moving objects database. In this paper, we propose an index framework, the UTR-Tree, for indexing the full uncertain trajectories of network constrained moving objects. Through a dynamic index maintenance technique which is associated with location updates, the UTR-Tree can deal with the full uncertain trajectories, which include not only the historical locations of moving objects, but also their current and near future location information with uncertainty considered, so that the queries on the whole life span of the moving objects can be efficiently supported. The experimental results show that the UTR-Tree outperforms previously proposed network-based moving object index methods in dealing with full uncertain trajectories.