Unsplit/Dissolve Multiple Lines in a Stream Network

I am trying to dissolve multiple lines for stream networks into individual features. The dataset contains many stream networks so a complete dissolve of the whole dataset wont do as I would like to only dissolve streams that are in the same network (illustrated in the pictures below).

I've tried using the Unsplit Line tool and the Dissolve tool in ArcGIS 10. The tool will only join two lines together, i.e. lines that share a common end point. This is illustrated in the screenshots below.

Here is the orignal dataset with the lines split, a single line feature is highlighted

This is the same feature highlighted after using the unsplit/dissolve tool. As you can see it has only joined two lines together rather than all of the lines that are touching.

All suggestions welcomed. I am using ArcGIS 10, but am open to using QGIS (although from what I can tell the dissolve tool in QGIS does the same).

Thank You