How can I recreate a directory structure, without the files?
With the cpio program:
cd "$srcdir" find . -type d -print | cpio -pdumv "$dstdir"
or with GNU tar, and more verbose syntax:
cd "$srcdir" find . -type d -print | tar c --files-from - --no-recursion | tar x --directory "$dstdir"
This creates a list of directory names with find, non-recursively adds just the directories to an archive, and pipes it to a second tar instance to extract it at the target location.