diff --git a/asdf/search.py b/asdf/search.py
index d8507c5d8..531183d45 100644
--- a/asdf/search.py
+++ b/asdf/search.py
@@ -6,14 +6,21 @@
 import re
 import typing
 
+from . import _itertree
 from ._display import DEFAULT_MAX_COLS, DEFAULT_MAX_ROWS, DEFAULT_SHOW_VALUES, format_faint, format_italic, render_tree
 from ._node_info import NodeSchemaInfo, collect_schema_info
-from .treeutil import _get_children, _is_container
+from .treeutil import _is_container
 from .util import NotSet
 
 __all__ = ["AsdfSearchResult"]
 
 
+def _get_children(obj):
+    if hasattr(obj, "__asdf_traverse__"):
+        obj = obj.__asdf_traverse__()
+    return _itertree._default_get_children(obj)
+
+
 class AsdfSearchResult:
     """
     Result of a call to AsdfFile.search.
@@ -222,11 +229,9 @@ def replace(self, value):
         """
         results = []
 
-        def _callback(identifiers, parent, node, children):
-            if all(f(node, identifiers[-1]) for f in self._filters):
-                results.append((identifiers[-1], parent))
-
-        _walk_tree_breadth_first(self._identifiers, self._node, _callback)
+        for node, edge in _itertree.breadth_first(self._node, get_children=_get_children):
+            if all(f(node, edge.key) for f in self._filters):
+                results.append((edge.key, edge.parent.node))
 
         for identifier, parent in results:
             parent[identifier] = value
@@ -284,11 +289,10 @@ def nodes(self):
         """
         results = []
 
-        def _callback(identifiers, parent, node, children):
-            if all(f(node, identifiers[-1]) for f in self._filters):
+        for node, edge in _itertree.breadth_first(self._node, get_children=_get_children):
+            if all(f(node, edge.key) for f in self._filters):
                 results.append(node)
 
-        _walk_tree_breadth_first(self._identifiers, self._node, _callback)
         return results
 
     @property
@@ -303,11 +307,10 @@ def paths(self):
         """
         results = []
 
-        def _callback(identifiers, parent, node, children):
-            if all(f(node, identifiers[-1]) for f in self._filters):
-                results.append(_build_path(identifiers))
+        for node, edge in _itertree.breadth_first(self._node, get_children=_get_children):
+            if all(f(node, edge.key) for f in self._filters):
+                results.append(_build_path(self._identifiers + list(_itertree.edge_to_keys(edge))))
 
-        _walk_tree_breadth_first(self._identifiers, self._node, _callback)
         return results
 
     def __repr__(self):
@@ -369,31 +372,6 @@ def __getitem__(self, key):
         )
 
 
-def _walk_tree_breadth_first(root_identifiers, root_node, callback):
-    """
-    Walk the tree in breadth-first order (useful for prioritizing
-    lower-depth nodes).
-    """
-    current_nodes = [(root_identifiers, None, root_node)]
-    seen = set()
-    while True:
-        next_nodes = []
-
-        for identifiers, parent, node in current_nodes:
-            if (isinstance(node, (dict, list, tuple)) or NodeSchemaInfo.traversable(node)) and id(node) in seen:
-                continue
-            tnode = node.__asdf_traverse__() if NodeSchemaInfo.traversable(node) else node
-            children = _get_children(tnode)
-            callback(identifiers, parent, node, [c for _, c in children])
-            next_nodes.extend([([*identifiers, i], node, c) for i, c in children])
-            seen.add(id(node))
-
-        if len(next_nodes) == 0:
-            break
-
-        current_nodes = next_nodes
-
-
 def _build_path(identifiers):
     """
     Generate the Python code needed to extract the identified node.