Unfortunately, we don't have a way of pulling a full project hierarchy for a specific project using the API. It would be possible to build a script to do this though.
You would start at the top-level project and pull its direct children initially with the following (replacing the project ObjectID I have highlighted in yellow with your project information):
When inspecting the output of the above query, for every project that has "Children" count > 0, it would then be necessary to follow that query down into those child projects etc.....
Depending on the use case, It would also be possible for a Rally Subscription Administrator to drop this information into an Excel spreadsheet by using the Administrators pages to expand all the projects and then copy and paste the project data into an excel spreadsheet. The Admin could also narrow this down by selecting the top project you are wanting and then selecting the children of that project before copying and pasting.