Directory Size Aggregation

Consider a file system represented as a nested dictionary structure. In this structure:

• A key represents the name of either a file or a directory.
• For files, the value is an integer representing its size (in bytes).
• For directories, the value is itself a dictionary following the same format.

For example:

{
  "file1.txt": 100,
  "folder1": {
    "file2.txt": 200,
    "folder2": {
      "file3.txt": 300
    }
  }
}

Write a function that takes such a file system dictionary as input and returns a dictionary where:

• Each key is the full path of a directory.
• Each value is the total size of all files contained within that directory and all of its subdirectories.

The root directory should be represented by /, and the directories nested inside should be represented by their full paths (e.g., /folder1, /folder1/folder2).

Your function should correctly handle arbitrarily nested directories.