[MT-TREE] Support prediction cache and model slicing. (#8968)

- Fix prediction range.
- Support prediction cache in mt-hist.
- Support model slicing.
- Make the booster a Python iterable by defining `__iter__`.
- Cleanup removed/deprecated parameters.
- A new field in the output model `iteration_indptr` for pointing to the ranges of trees for each iteration.

tests/python/test_basic_models.py

@@ -524,7 +524,7 @@ class TestModels:
             booster[-1:0]
 
         # we do not accept empty slice.
-        with pytest.raises(ValueError):
+        with pytest.raises(ValueError, match="Empty slice"):
             booster[1:1]
         # stop can not be smaller than begin
         with pytest.raises(ValueError, match=r"Invalid.*"):
@@ -615,6 +615,46 @@ class TestModels:
         booster = xgb.Booster(model_file=bytesarray)
         self.run_slice(booster, dtrain, num_parallel_tree, num_classes, num_boost_round)
 
+    def test_slice_multi(self) -> None:
+        from sklearn.datasets import make_classification
+
+        num_classes = 3
+        X, y = make_classification(
+            n_samples=1000, n_informative=5, n_classes=num_classes
+        )
+        Xy = xgb.DMatrix(data=X, label=y)
+        num_parallel_tree = 4
+        num_boost_round = 16
+
+        class ResetStrategy(xgb.callback.TrainingCallback):
+            def after_iteration(self, model, epoch: int, evals_log) -> bool:
+                model.set_param({"multi_strategy": "multi_output_tree"})
+                return False
+
+        booster = xgb.train(
+            {
+                "num_parallel_tree": num_parallel_tree,
+                "num_class": num_classes,
+                "booster": "gbtree",
+                "objective": "multi:softprob",
+                "multi_strategy": "multi_output_tree",
+                "tree_method": "hist",
+                "base_score": 0,
+            },
+            num_boost_round=num_boost_round,
+            dtrain=Xy,
+            callbacks=[ResetStrategy()]
+        )
+        sliced = [t for t in booster]
+        assert len(sliced) == 16
+
+        predt0 = booster.predict(Xy, output_margin=True)
+        predt1 = np.zeros(predt0.shape)
+        for t in booster:
+            predt1 += t.predict(Xy, output_margin=True)
+
+        np.testing.assert_allclose(predt0, predt1, atol=1e-5)
+
     @pytest.mark.skipif(**tm.no_pandas())
     def test_feature_info(self):
         import pandas as pd