I have a treeview that I can build inside of a project and I would like it to be saved with the current ArcGIS project, so when I open the project and activate the dockpane it re-hydrates the saved contents. Is this ability available and if so what objects do I need to use or are their some examples I can reference?
Custom project settings can be saved within a Pro project and restored when you reopen the project. Here is a guide that can help you with how you can create your own custom project settings using the Pro SDK.
There is a sample that explains the complete workflow explained in this guide: BackStage_PropertyPage