By default, the exports done through File > Export contain only folder and subfolders. To include also the files, click the "Customize" button In the Save-dialog and click "Include single files in export" and also "Export the full directory branch" if applicable. You will find these options also at: "Tools > Options > Export".
Using "File > Export > Copy List of Files" you can easily copy a list of files to the clipboard. This function is also included in the context menu of the TreeSize directory tree.