diff --git a/networkx/algorithms/approximation/steinertree.py b/networkx/algorithms/approximation/steinertree.py
index c6c834f422c..475bea6890d 100644
--- a/networkx/algorithms/approximation/steinertree.py
+++ b/networkx/algorithms/approximation/steinertree.py
@@ -114,9 +114,18 @@ def _kou_steiner_tree(G, terminal_nodes, weight):
 
 def _remove_nonterminal_leaves(G, terminals):
     terminals_set = set(terminals)
-    for n in list(G.nodes):
-        if n not in terminals_set and G.degree(n) == 1:
-            G.remove_node(n)
+    degree_one_nodes = {n for n in G if G.degree(n) == 1}
+    nonterminal_leaves = degree_one_nodes - terminals_set
+    while nonterminal_leaves:
+        possible_nonterminal_leaves = set()
+        for n in nonterminal_leaves:
+            possible_nonterminal_leaves = possible_nonterminal_leaves | set(
+                G.neighbors(n)
+            )
+        G.remove_nodes_from(nonterminal_leaves)
+        nonterminal_leaves = {
+            leaf for leaf in possible_nonterminal_leaves if G.degree(leaf) == 1
+        } - terminals_set
 
 
 ALGORITHMS = {
diff --git a/networkx/algorithms/approximation/tests/test_steinertree.py b/networkx/algorithms/approximation/tests/test_steinertree.py
index 23c3193e42e..763a9618f3c 100644
--- a/networkx/algorithms/approximation/tests/test_steinertree.py
+++ b/networkx/algorithms/approximation/tests/test_steinertree.py
@@ -190,6 +190,12 @@ def test_multigraph_steiner_tree(self):
             S = steiner_tree(G, terminal_nodes, method=method)
             assert edges_equal(S.edges(data=True, keys=True), expected_edges)
 
+    def test_remove_nonterminal_leaves(self):
+        G = nx.path_graph(10)
+        nx.approximation.steinertree._remove_nonterminal_leaves(G, [4, 5, 6])
+
+        assert list(G) == [4, 5, 6]  # only the terminal nodes are left
+
 
 @pytest.mark.parametrize("method", ("kou", "mehlhorn"))
 def test_steiner_tree_weight_attribute(method):