Datatree deduplicate setitem (#9602)

* test

* bugfix

Author: TomNicholas

xarray/core/datatree.py

@@ -813,6 +813,9 @@ def _replace_node(
         if data is not _default:
             self._set_node_data(ds)
 
+        if self.parent is not None:
+            _deduplicate_inherited_coordinates(self, self.parent)
+
         self.children = children
 
     def _copy_node(

xarray/tests/test_datatree.py

@@ -1315,6 +1315,19 @@ def test_inherited_coords_with_index_are_deduplicated(self):
         expected = xr.Dataset({"foo": ("x", [4, 5])})
         assert_identical(child_dataset, expected)
 
+    def test_deduplicated_after_setitem(self):
+        # regression test for GH #9601
+        dt = DataTree.from_dict(
+            {
+                "/": xr.Dataset(coords={"x": [1, 2]}),
+                "/b": None,
+            }
+        )
+        dt["b/x"] = dt["x"]
+        child_dataset = dt.children["b"].to_dataset(inherited=False)
+        expected = xr.Dataset()
+        assert_identical(child_dataset, expected)
+
     def test_inconsistent_dims(self):
         expected_msg = _exact_match(
             """