diff --git a/lib/spack/spack/spec.py b/lib/spack/spack/spec.py
index bc209f92fde..f6bcaf5835a 100644
--- a/lib/spack/spack/spec.py
+++ b/lib/spack/spack/spec.py
@@ -3815,8 +3815,7 @@ def _cmp_iter(self):
         #   2. A generator over canonical edges
         #
         # Canonical edges have consistent ids defined by breadth-first traversal order. That is,
-        # the fake edge from nothing to the root is (0, 1), the root is always 1, dependencies of
-        # the root are 2, 3, 4, 5, etc., and so on.
+        # the root is always 0, dependencies of the root are 1, 2, 3, etc., and so on.
         #
         # The three cases are:
         #
@@ -3839,19 +3838,14 @@ def _cmp_iter(self):
         # need for the complexity here. It was not clear at the time of writing that how
         # much optimization was possible in `spack.traverse`.
 
-        def yield_root_edge():
-            yield 0  # id of None (parent of root)
-            yield 1  # id of root
-            yield 0  # depflag on fake edge
-            yield ()  # virtuals on fake edge
-
         if not self._dependencies:
             # Spec has no dependencies -- simplest case
             def nodes():
                 yield self._cmp_node  # just yield this node
 
             def edges():
-                yield yield_root_edge
+                return
+                yield
 
         elif not any(
             dep.spec._dependencies for deplist in self._dependencies.values() for dep in deplist
@@ -3879,15 +3873,12 @@ def nodes():
                     yield edge.spec._cmp_node
 
             def edges():
-                # yield root edge consistent with what traverse would do.
-                yield yield_root_edge
-
                 # we've already sorted to yield dependency nodes in order, now yield
                 # edges for comparison in the same order, with BFS-consistent ids
-                for i, edge in enumerate(sorted_edges, start=2):
+                for i, edge in enumerate(sorted_edges, start=1):
 
                     def yield_edge(i=i, edge=edge):
-                        yield 1  # id of root
+                        yield 0  # id of root
                         yield i  # id of dependency in BFS order
                         yield edge.depflag
                         yield edge.virtuals
@@ -3900,7 +3891,6 @@ def yield_edge(i=i, edge=edge):
             # We need a way to compare edges consistently between two arbitrary specs.
             # node_ids generates consistent node ids based on BFS traversal order.
             node_ids = collections.defaultdict(lambda: len(node_ids))
-            node_ids[id(None)]  # None (parent of root) is always node id 0
 
             # To avoid traversing twice, we put edges in this list as we yield the nodes.
             # edges() yields the edge functions later, if necessary.
@@ -3915,6 +3905,10 @@ def nodes():
                         yield edge.spec._cmp_node
                         node_ids[id(edge.spec)]
 
+                    # skip fake edge to root
+                    if edge.parent is None:
+                        continue
+
                     # create "edges" with the ids we generated above
                     def yield_edge(edge=edge):
                         yield node_ids[id(edge.parent)]
diff --git a/lib/spack/spack/test/spec_semantics.py b/lib/spack/spack/test/spec_semantics.py
index 7c995796aa7..29dd072506d 100644
--- a/lib/spack/spack/test/spec_semantics.py
+++ b/lib/spack/spack/test/spec_semantics.py
@@ -2057,10 +2057,7 @@ def test_spec_ordering(specs_in_expected_order):
     "spec,expected_tuplified",
     [
         # simple, no dependencies
-        [
-            ("a"),
-            ((("a", None, EMPTY_VER, EMPTY_VAR, EMPTY_FLG, None, None, None),), ((0, 1, 0, ()),)),
-        ],
+        [("a"), ((("a", None, EMPTY_VER, EMPTY_VAR, EMPTY_FLG, None, None, None),), ())],
         # with some node attributes
         [
             ("a@1.0 +foo cflags='-O3 -g'"),
@@ -2077,7 +2074,7 @@ def test_spec_ordering(specs_in_expected_order):
                         None,
                     ),
                 ),
-                ((0, 1, 0, ()),),
+                (),
             ),
         ],
         # single edge case
@@ -2088,7 +2085,7 @@ def test_spec_ordering(specs_in_expected_order):
                     ("a", None, EMPTY_VER, EMPTY_VAR, EMPTY_FLG, None, None, None),
                     ("b", None, EMPTY_VER, EMPTY_VAR, EMPTY_FLG, None, None, None),
                 ),
-                ((0, 1, 0, ()), (1, 2, 0, ())),
+                ((0, 1, 0, ()),),
             ),
         ],
         # root with multiple deps
@@ -2101,7 +2098,7 @@ def test_spec_ordering(specs_in_expected_order):
                     ("c", None, EMPTY_VER, EMPTY_VAR, EMPTY_FLG, None, None, None),
                     ("d", None, EMPTY_VER, EMPTY_VAR, EMPTY_FLG, None, None, None),
                 ),
-                ((0, 1, 0, ()), (1, 2, 0, ()), (1, 3, 0, ()), (1, 4, 0, ())),
+                ((0, 1, 0, ()), (0, 2, 0, ()), (0, 3, 0, ())),
             ),
         ],
         # root with multiple build deps
@@ -2114,7 +2111,7 @@ def test_spec_ordering(specs_in_expected_order):
                     ("c", None, EMPTY_VER, EMPTY_VAR, EMPTY_FLG, None, None, None),
                     ("d", None, EMPTY_VER, EMPTY_VAR, EMPTY_FLG, None, None, None),
                 ),
-                ((0, 1, 0, ()), (1, 2, dt.BUILD, ()), (1, 3, dt.BUILD, ()), (1, 4, dt.BUILD, ())),
+                ((0, 1, dt.BUILD, ()), (0, 2, dt.BUILD, ()), (0, 3, dt.BUILD, ())),
             ),
         ],
         # dependencies with dependencies
@@ -2132,12 +2129,11 @@ def test_spec_ordering(specs_in_expected_order):
                 ),
                 (
                     (0, 1, 0, ()),
-                    (1, 2, 0, ()),
-                    (1, 3, 0, ()),
-                    (2, 4, dt.BUILD, ()),
+                    (0, 2, 0, ()),
+                    (1, 3, dt.BUILD, ()),
+                    (1, 4, dt.BUILD, ()),
                     (2, 5, dt.BUILD, ()),
-                    (3, 6, dt.BUILD, ()),
-                    (3, 7, dt.BUILD, ()),
+                    (2, 6, dt.BUILD, ()),
                 ),
             ),
         ],