import os
 
def get_depth(file_or_dir_path):
    abs_path = os.path.abspath(os.path.expanduser(file_or_dir_path))
    depth = os.path.normpath(abs_path).count(os.sep)
    return depth

Examples:

In [2]: 
os.path.abspath(os.path.expanduser('~'))
Out[2]: 
'/home/rajulocal'

In [3]: 
[get_depth('~'), get_depth('/home/rajulocal'), get_depth('/home/rajulocal/'),
 get_depth('/home/rajulocal/foo'), get_depth('/'), get_depth('/home')]
Out[3]: 
[2, 2, 2, 3, 1, 1]

Using Python 3.10.6, Ipython 8.6.0

tags | measure the depth of a file system path

See also:

• https://stackoverflow.com/questions/31528199/how-to-measure-the-depth-of-a-file-system-path - shows how to do it with various command line tools.
• https://docs.python.org/3/library/os.path.html

tags | relative path

import os
 
def dir_depth_relative(child, parent):
    child = os.path.abspath(os.path.expanduser(child))
    parent = os.path.abspath(os.path.expanduser(parent))
    assert child.startswith(parent), "child parent relation violated. parent = " + parent + ', child = ' + child
    name_space = os.path.relpath(child, parent)
    level = os.path.normpath(name_space).count(os.sep) + 1 if name_space != '.' else 0
    return level

Examples:

In [2]: 
[dir_depth_relative('~', '~'), dir_depth_relative('~', '~/'),
 dir_depth_relative('~/work', '~'), dir_depth_relative('~/work.txt', '~'),
 dir_depth_relative('/tmp/foo', '/tmp'), dir_depth_relative('/tmp', '/tmp'),
 dir_depth_relative('/tmp/foo/bar', '/tmp')]
Out[2]: 
[0, 0, 1, 1, 1, 0, 2]

In [3]: 
dir_depth_relative('/tmp', '/tmp/foo')
---------------------------------------------------------------------------
AssertionError                            Traceback (most recent call last)
Cell In [3], line 1
----> 1 dir_depth_relative('/tmp', '/tmp/foo')

Cell In [1], line 6, in dir_depth_relative(child, parent)
      4 child = os.path.abspath(os.path.expanduser(child))
      5 parent = os.path.abspath(os.path.expanduser(parent))
----> 6 assert child.startswith(parent), "child parent relation violated. parent = " + parent + ', child = ' + child
      7 name_space = os.path.relpath(child, parent)
      8 level = os.path.normpath(name_space).count(os.sep) + 1 if name_space != '.' else 0

AssertionError: child parent relation violated. parent = /tmp/foo, child = /tmp