BUG: Fix factorize to ensure proper use of null_encoding parameter an…

…d backwards compatibility maintained

pandas/core/arrays/arrow/array.py

@@ -1207,9 +1207,15 @@ def factorize(
             # https://github.com/apache/arrow/issues/15226#issuecomment-1376578323
             data = data.cast(pa.int64())
 
-        if pa.types.is_dictionary(data.type) and null_encoding == "encode":
-            data = data.cast(data.type.value_type)
-        encoded = data.dictionary_encode(null_encoding=null_encoding)
+        if pa.types.is_dictionary(data.type):
+            if null_encoding == "encode":
+                # dictionary encode does nothing if an already encoded array is given
+                data = data.cast(data.type.value_type)
+                encoded = data.dictionary_encode(null_encoding=null_encoding)
+            else:
+                encoded = data
+        else:
+            encoded = data.dictionary_encode(null_encoding=null_encoding)
         if encoded.length() == 0:
             indices = np.array([], dtype=np.intp)
             uniques = type(self)(pa.chunked_array([], type=encoded.type.value_type))