diff --git a/optax/projections/_projections_test.py b/optax/projections/_projections_test.py
index 2f4ea6a48..00e142b48 100644
--- a/optax/projections/_projections_test.py
+++ b/optax/projections/_projections_test.py
@@ -50,7 +50,7 @@ def setUp(self):
         'l1': (proj.projection_l1_ball, partial(optax.tree.norm, ord=1)),
         'l2': (proj.projection_l2_ball, optax.tree.norm),
         'linf': (proj.projection_linf_ball,
-                 partial(optax.tree.norm, ord='inf')),
+                 partial(optax.tree.norm, ord=float('inf'))),
     }
 
   def test_projection_non_negative(self):
diff --git a/optax/tree_utils/__init__.py b/optax/tree_utils/__init__.py
index ce4952f2d..96324d0ba 100644
--- a/optax/tree_utils/__init__.py
+++ b/optax/tree_utils/__init__.py
@@ -83,7 +83,7 @@
         ('optax.tree_utils.tree_linf_norm is deprecated: use'
          ' optax.tree_utils.tree_norm(..., ord=jnp.inf)'
          ' (optax v0.2.5 or newer).'),
-        functools.partial(tree_norm, ord='inf'),
+        functools.partial(tree_norm, ord=float('inf')),
     ),
 }
 
@@ -94,7 +94,7 @@
   tree_add_scalar_mul = tree_add_scale
   tree_l1_norm = functools.partial(tree_norm, ord=1)
   tree_l2_norm = tree_norm
-  tree_linf_norm = functools.partial(tree_norm, ord='inf')
+  tree_linf_norm = functools.partial(tree_norm, ord=float('inf'))
 
 else:
   # pylint: disable=line-too-long
diff --git a/optax/tree_utils/_tree_math.py b/optax/tree_utils/_tree_math.py
index 9387a0976..67d2ed4f6 100644
--- a/optax/tree_utils/_tree_math.py
+++ b/optax/tree_utils/_tree_math.py
@@ -236,25 +236,25 @@ def _square(leaf):
 
 
 def tree_norm(tree: Any,
-              ord: int | str | float | None = None,  # pylint: disable=redefined-builtin
+              ord: int | float = 2,  # pylint: disable=redefined-builtin
               squared: bool = False) -> jax.Array:
   """Compute the vector norm of the given ord of a pytree.
 
   Args:
     tree: pytree.
-    ord: the order of the vector norm to compute from (None, 1, 2, inf).
+    ord: the order of the vector norm to compute, one of ``{1, 2, inf}``.
     squared: whether the norm should be returned squared or not.
 
   Returns:
     a scalar value.
   """
-  if ord is None or ord == 2:
+  if ord == 2:
     squared_tree = jax.tree.map(_square, tree)
     sqnorm = tree_sum(squared_tree)
     return jnp.array(sqnorm if squared else jnp.sqrt(sqnorm))
   elif ord == 1:
     ret = tree_sum(jax.tree.map(jnp.abs, tree))
-  elif ord == jnp.inf or ord in ("inf", "infinity"):
+  elif ord == float("inf"):
     ret = tree_max(jax.tree.map(jnp.abs, tree))
   else:
     raise ValueError(f"Unsupported ord: {ord}")