diff --git a/Octree/include/CGAL/Octree/IO.h b/Octree/include/CGAL/Octree/IO.h
index 3ae74b90898..5ea3f95340c 100644
--- a/Octree/include/CGAL/Octree/IO.h
+++ b/Octree/include/CGAL/Octree/IO.h
@@ -17,7 +17,7 @@ template<class PointRange,
 ostream &operator<<(ostream &os, const CGAL::Octree::Octree<PointRange, PointMap> &octree) {
 
   // Create a range of nodes
-  auto nodes = octree.template walk<CGAL::Octree::Walker::Preorder_tree_walker>();
+  auto nodes = octree.template walk<CGAL::Octree::Walker::Preorder>();
 
   // Iterate over the range and print each node
   for (auto &n : nodes)
diff --git a/Octree/include/CGAL/Octree/Walker_criterion.h b/Octree/include/CGAL/Octree/Walker_criterion.h
index dca931d21e6..abfd28d5b07 100644
--- a/Octree/include/CGAL/Octree/Walker_criterion.h
+++ b/Octree/include/CGAL/Octree/Walker_criterion.h
@@ -75,7 +75,7 @@ struct Preorder {
   }
 
   template<class Value>
-  const Node::Node <Value> *operator()(const Node::Node <Value> *n) const {
+  const Node::Node <Value> *next(const Node::Node <Value> *n) const {
 
     if (n->is_leaf()) {
 
@@ -100,13 +100,13 @@ struct Preorder {
 struct Postorder {
 
   template<class Value>
-  const Node::Node <Value> *first(const Node::Node <Value> *root) {
+  const Node::Node <Value> *first(const Node::Node <Value> *root) const {
 
     return deepest_first_child(root);
   }
 
   template<class Value>
-  const Node::Node <Value> *operator()(const Node::Node <Value> *n) {
+  const Node::Node <Value> *next(const Node::Node <Value> *n) const {
 
     auto next = deepest_first_child(next_sibling(n));
 
@@ -120,13 +120,13 @@ struct Postorder {
 struct Leaves {
 
   template<class Value>
-  const Node::Node <Value> *first(const Node::Node <Value> *root) {
+  const Node::Node <Value> *first(const Node::Node <Value> *root) const {
 
     return deepest_first_child(root);
   }
 
   template<class Value>
-  const Node::Node <Value> *operator()(const Node::Node <Value> *n) {
+  const Node::Node <Value> *next(const Node::Node <Value> *n) const {
 
     auto next = deepest_first_child(next_sibling(n));
 
@@ -137,38 +137,6 @@ struct Leaves {
   }
 };
 
-class Preorder_tree_walker {
-
-public:
-
-  template<class Value>
-  const Node::Node <Value> *first(const Node::Node <Value> *root) const {
-    return root;
-  }
-
-  template<class Value>
-  const Node::Node <Value> *next(const Node::Node <Value> *n) const {
-
-    if (n->is_leaf()) {
-
-      auto next = next_sibling(n);
-
-      if (nullptr == next) {
-
-        return next_sibling_up(n);
-      }
-
-      return next;
-
-    } else {
-
-      return &(*n)[0];
-    }
-
-  }
-};
-
-
 }
 
 }
diff --git a/Octree/test/Octree/test_tree_walk.cpp b/Octree/test/Octree/test_tree_walk.cpp
index 05973050f8a..93cf8f5abae 100644
--- a/Octree/test/Octree/test_tree_walk.cpp
+++ b/Octree/test/Octree/test_tree_walk.cpp
@@ -28,7 +28,7 @@ bool test_preorder_1_node() {
   octree.refine(10, 1);
 
   // Create the range
-  auto nodes = octree.walk<CGAL::Octree::Walker::Preorder_tree_walker>();
+  auto nodes = octree.walk<CGAL::Octree::Walker::Preorder>();
 
   // Check each item in the range
   auto iter = nodes.begin();
@@ -50,7 +50,7 @@ bool test_preorder_9_nodes() {
   octree.refine(10, 1);
 
   // Create the range
-  auto nodes = octree.walk<CGAL::Octree::Walker::Preorder_tree_walker>();
+  auto nodes = octree.walk<CGAL::Octree::Walker::Preorder>();
 
   // Check each item in the range
   auto iter = nodes.begin();
@@ -78,7 +78,7 @@ bool test_preorder_25_nodes() {
   octree.refine(10, 1);
 
   // Create the range
-  auto nodes = octree.walk<CGAL::Octree::Walker::Preorder_tree_walker>();
+  auto nodes = octree.walk<CGAL::Octree::Walker::Preorder>();
 
   // Check each item in the range
   auto iter = nodes.begin();