Maximum depth of tree

Given the root of a binary tree, write a function `max_depth(root)` that returns the maximum depth (height) of the tree. The maximum depth is the number of nodes along the longest path from the root node down to the farthest leaf node. A binary tree node is represented as a dictionary with keys `val`, `left`, and `right`. For example, a node with value 3, left child 9, and right child 20 would be `{'val': 3, 'left': {'val': 9, 'left': None, 'right': None}, 'right': {'val': 20, 'left': None, 'right': None}}`. The `left` and `right` values are either `None` (no child) or another dictionary with the same structure. If the tree is empty (`root is None`), the depth is `0`. The root node itself counts as depth `1`. Your task: implement `max_depth(root)` that takes such a dictionary-based root (or `None`) and returns the maximum depth as an integer.

>>> # Example 1: tree = [3,9,20,None,None,15,7]
>>> root = {'val': 3, 'left': {'val': 9, 'left': None, 'right': None}, 'right': {'val': 20, 'left': {'val': 15, 'left': None, 'right': None}, 'right': {'val': 7, 'left': None, 'right': None}}}
>>> max_depth(root)
3

>>> # Example 2: empty tree
>>> max_depth(None)
0

>>> # Example 3: single node
>>> max_depth({'val': 1, 'left': None, 'right': None})
1