diff --git a/Orthtree/include/CGAL/Orthtree.h b/Orthtree/include/CGAL/Orthtree.h
index 4ee5f787650..dce5637e089 100644
--- a/Orthtree/include/CGAL/Orthtree.h
+++ b/Orthtree/include/CGAL/Orthtree.h
@@ -247,7 +247,7 @@ public:
     // save orthtree attributes
     m_bbox_min = construct_point_d_from_array(bbox_min);
     m_side_per_depth.push_back(bbox_max[0] - bbox_min[0]);
-    points(index(root())) = {point_range.begin(), point_range.end()};
+    points(root()) = {point_range.begin(), point_range.end()};
   }
 
   /// @}
@@ -436,15 +436,8 @@ public:
 
     \return a const reference to the root node of the tree.
    */
-  const Node& root() const { return m_nodes[0]; }
-
-  /*!
-    \brief provides read-write access to the root node, and by
-    extension the rest of the tree.
-
-    \return a reference to the root node of the tree.
-   */
-  Node& root() { return m_nodes[0]; }
+  // todo: return index instead of ref
+  Node_index root() const { return 0; }
 
   Node_index index(const Node& node) const {
     return std::distance(m_nodes.data(), &node);
@@ -571,7 +564,7 @@ public:
     CGAL_precondition (CGAL::do_intersect(point, bbox(root())));
 
     // Start at the root node
-    Node_index node_for_point = index(root());
+    Node_index node_for_point = root();
 
     // Descend the tree until reaching a leaf node
     while (!is_leaf(node_for_point)) {
@@ -643,7 +636,7 @@ public:
    */
   template <typename Query, typename OutputIterator>
   OutputIterator intersected_nodes(const Query& query, OutputIterator output) const {
-    return intersected_nodes_recursive(query, index(root()), output);
+    return intersected_nodes_recursive(query, root(), output);
   }
 
   /// @}
@@ -716,16 +709,6 @@ public:
     \brief returns this node's parent.
     \pre `!is_root()`
    */
-  const Node& parent(const Node& node) const {
-    CGAL_precondition (!node.is_root());
-    return m_nodes[node.m_parent_index.get()];
-  }
-
-  Node& parent(Node& node) {
-    CGAL_precondition (!node.is_root());
-    return m_nodes[node.m_parent_index.get()];
-  }
-
   Node_index parent(Node_index node) const {
     CGAL_precondition (!is_root(node));
     return m_nodes[node].m_parent_index.get();
@@ -736,10 +719,6 @@ public:
     return m_nodes[node].m_children_index.get() + i;
   }
 
-  // todo: these types can probably be moved out of Node
-  using Children = typename Node::Children;
-  using Children_const = typename Node::Children_const;
-
   const boost::optional<Node_index> next_sibling(Node_index n) const {
 
     // Root node has no siblings
@@ -883,7 +862,7 @@ public:
   }
 
   static bool is_topology_equal(const Self& lhs, const Self& rhs) {
-    return is_topology_equal(lhs.index(lhs.root()), lhs, rhs.index(rhs.root()), rhs);
+    return is_topology_equal(lhs.root(), lhs, rhs.root(), rhs);
   }
 
   /*!
@@ -1176,7 +1155,7 @@ private: // functions :
       points_list.reserve(k);
 
     // Invoking the recursive function adds those points to the vector (passed by reference)
-    nearest_k_neighbors_recursive(query_sphere, index(root()), points_list);
+    nearest_k_neighbors_recursive(query_sphere, root(), points_list);
 
     // Add all the points found to the output
     for (auto& item: points_list)
diff --git a/Orthtree/include/CGAL/Orthtree/Traversals.h b/Orthtree/include/CGAL/Orthtree/Traversals.h
index ab224840442..3c5cf2dc43f 100644
--- a/Orthtree/include/CGAL/Orthtree/Traversals.h
+++ b/Orthtree/include/CGAL/Orthtree/Traversals.h
@@ -52,11 +52,11 @@ public:
   Preorder_traversal(const Tree& orthtree) : m_orthtree(orthtree) {}
 
   const Node* first() const {
-    return &m_orthtree.root();
+    return &m_orthtree[m_orthtree.root()];
   }
 
   typename Tree::Node_index first_index() const {
-    return m_orthtree.index(first()).get();
+    return m_orthtree.root();
   }
 
   const Node* next(const Node* n) const {
@@ -210,7 +210,7 @@ public:
 
   typename Tree::Node_index first_index() const {
     // assumes the tree has at least one child at m_depth
-    return m_orthtree.first_child_at_depth(m_orthtree.index(m_orthtree.root()), m_depth).get();
+    return m_orthtree.first_child_at_depth(m_orthtree.root(), m_depth).get();
   }
 
   template <typename Node>
diff --git a/Orthtree/test/Orthtree/test_node_adjacent.cpp b/Orthtree/test/Orthtree/test_node_adjacent.cpp
index cb248b0425a..a150f458121 100644
--- a/Orthtree/test/Orthtree/test_node_adjacent.cpp
+++ b/Orthtree/test/Orthtree/test_node_adjacent.cpp
@@ -42,37 +42,37 @@ int main(void) {
   std::cout << octree << std::endl;
 
   // Root node should have no siblings
-  assert(!octree.adjacent_node(octree.index(octree.root()), 0));
-  assert(!octree.adjacent_node(octree.index(octree.root()), 1));
-  assert(!octree.adjacent_node(octree.index(octree.root()), 2));
-  assert(!octree.adjacent_node(octree.index(octree.root()), 3));
-  assert(!octree.adjacent_node(octree.index(octree.root()), 4));
-  assert(!octree.adjacent_node(octree.index(octree.root()), 5));
+  assert(!octree.adjacent_node(octree.root(), 0));
+  assert(!octree.adjacent_node(octree.root(), 1));
+  assert(!octree.adjacent_node(octree.root(), 2));
+  assert(!octree.adjacent_node(octree.root(), 3));
+  assert(!octree.adjacent_node(octree.root(), 4));
+  assert(!octree.adjacent_node(octree.root(), 5));
 
   // Left Top Front node should have siblings to the Right, Down, and Back
-  auto left_top_back = octree.child(octree.index(octree.root()), Traits::LEFT_TOP_BACK);
+  auto left_top_back = octree.child(octree.root(), Traits::LEFT_TOP_BACK);
 
-  assert(octree.child(octree.index(octree.root()), Traits::RIGHT_TOP_BACK) ==
+  assert(octree.child(octree.root(), Traits::RIGHT_TOP_BACK) ==
          octree.adjacent_node(left_top_back, Traits::RIGHT).get());
-  assert(octree.child(octree.index(octree.root()), Traits::LEFT_BOTTOM_BACK) ==
+  assert(octree.child(octree.root(), Traits::LEFT_BOTTOM_BACK) ==
          octree.adjacent_node(left_top_back, Traits::DOWN).get());
-  assert(octree.child(octree.index(octree.root()), Traits::LEFT_TOP_FRONT) ==
+  assert(octree.child(octree.root(), Traits::LEFT_TOP_FRONT) ==
          octree.adjacent_node(left_top_back, Traits::FRONT));
   assert(!octree.adjacent_node(left_top_back, Traits::LEFT));
   assert(!octree.adjacent_node(left_top_back, Traits::UP));
   assert(!octree.adjacent_node(left_top_back, Traits::BACK));
 
-  std::cout << octree[octree.child(octree.index(octree.root()), Traits::LEFT_BOTTOM_BACK)] << std::endl;
+  std::cout << octree[octree.child(octree.root(), Traits::LEFT_BOTTOM_BACK)] << std::endl;
 
   auto right_top_back_of_left_bottom_back =
-    octree.child(octree.child(octree.index(octree.root()), Traits::LEFT_BOTTOM_BACK), Traits::RIGHT_TOP_BACK);
+    octree.child(octree.child(octree.root(), Traits::LEFT_BOTTOM_BACK), Traits::RIGHT_TOP_BACK);
 
   assert(
-    octree.child(octree.child(octree.index(octree.root()), Traits::LEFT_BOTTOM_BACK), Traits::LEFT_TOP_BACK) ==
+    octree.child(octree.child(octree.root(), Traits::LEFT_BOTTOM_BACK), Traits::LEFT_TOP_BACK) ==
     octree.adjacent_node(right_top_back_of_left_bottom_back, Traits::LEFT)
   );
   assert(
-    octree.child(octree.index(octree.root()), Traits::RIGHT_BOTTOM_BACK) ==
+    octree.child(octree.root(), Traits::RIGHT_BOTTOM_BACK) ==
     octree.adjacent_node(right_top_back_of_left_bottom_back, Traits::RIGHT)
   );
   assert(octree.adjacent_node(right_top_back_of_left_bottom_back, Traits::RIGHT).has_value());
diff --git a/Orthtree/test/Orthtree/test_node_index.cpp b/Orthtree/test/Orthtree/test_node_index.cpp
index db4c8ca2170..34e6f24f5c6 100644
--- a/Orthtree/test/Orthtree/test_node_index.cpp
+++ b/Orthtree/test/Orthtree/test_node_index.cpp
@@ -37,11 +37,11 @@ int main(void) {
   Octree octree(points, points.point_map());
   octree.refine(10, 1);
 
-  std::cout << "root: " << octree.local_coordinates(octree.index(octree.root())) << std::endl;
-  std::cout << "first child: " << octree.local_coordinates(octree.child(octree.index(octree.root()), 0)) << std::endl;
-  std::cout << "fifth child: " << octree.local_coordinates(octree.child(octree.index(octree.root()), 4)) << std::endl;
+  std::cout << "root: " << octree.local_coordinates(octree.root()) << std::endl;
+  std::cout << "first child: " << octree.local_coordinates(octree.child(octree.root(), 0)) << std::endl;
+  std::cout << "fifth child: " << octree.local_coordinates(octree.child(octree.root(), 4)) << std::endl;
   std::cout << "fifth child of first child: "
-            << octree.local_coordinates(octree.child(octree.child(octree.index(octree.root()), 0), 4)) << std::endl;
+            << octree.local_coordinates(octree.child(octree.child(octree.root(), 0), 4)) << std::endl;
 
   // TODO
 
diff --git a/Orthtree/test/Orthtree/test_octree_bbox.cpp b/Orthtree/test/Orthtree/test_octree_bbox.cpp
index 4f3a4f3dabb..8c324de951a 100644
--- a/Orthtree/test/Orthtree/test_octree_bbox.cpp
+++ b/Orthtree/test/Orthtree/test_octree_bbox.cpp
@@ -42,14 +42,14 @@ void test_9_nodes() {
   assert(octree.bbox(octree.root()) == CGAL::Bbox_3(-1.1, -1.1, -1.1, 1.1, 1.1, 1.1));
 
   // Compare the child nodes
-  assert(octree.bbox(octree.child(octree.index(octree.root()), 0)) == CGAL::Bbox_3(-1.1, -1.1, -1.1, 0, 0, 0));
-  assert(octree.bbox(octree.child(octree.index(octree.root()), 1)) == CGAL::Bbox_3(0, -1.1, -1.1, 1.1, 0, 0));
-  assert(octree.bbox(octree.child(octree.index(octree.root()), 2)) == CGAL::Bbox_3(-1.1, 0, -1.1, 0, 1.1, 0));
-  assert(octree.bbox(octree.child(octree.index(octree.root()), 3)) == CGAL::Bbox_3(0, 0, -1.1, 1.1, 1.1, 0));
-  assert(octree.bbox(octree.child(octree.index(octree.root()), 4)) == CGAL::Bbox_3(-1.1, -1.1, 0, 0, 0, 1.1));
-  assert(octree.bbox(octree.child(octree.index(octree.root()), 5)) == CGAL::Bbox_3(0, -1.1, 0, 1.1, 0, 1.1));
-  assert(octree.bbox(octree.child(octree.index(octree.root()), 6)) == CGAL::Bbox_3(-1.1, 0, 0, 0, 1.1, 1.1));
-  assert(octree.bbox(octree.child(octree.index(octree.root()), 7)) == CGAL::Bbox_3(0, 0, 0, 1.1, 1.1, 1.1));
+  assert(octree.bbox(octree.child(octree.root(), 0)) == CGAL::Bbox_3(-1.1, -1.1, -1.1, 0, 0, 0));
+  assert(octree.bbox(octree.child(octree.root(), 1)) == CGAL::Bbox_3(0, -1.1, -1.1, 1.1, 0, 0));
+  assert(octree.bbox(octree.child(octree.root(), 2)) == CGAL::Bbox_3(-1.1, 0, -1.1, 0, 1.1, 0));
+  assert(octree.bbox(octree.child(octree.root(), 3)) == CGAL::Bbox_3(0, 0, -1.1, 1.1, 1.1, 0));
+  assert(octree.bbox(octree.child(octree.root(), 4)) == CGAL::Bbox_3(-1.1, -1.1, 0, 0, 0, 1.1));
+  assert(octree.bbox(octree.child(octree.root(), 5)) == CGAL::Bbox_3(0, -1.1, 0, 1.1, 0, 1.1));
+  assert(octree.bbox(octree.child(octree.root(), 6)) == CGAL::Bbox_3(-1.1, 0, 0, 0, 1.1, 1.1));
+  assert(octree.bbox(octree.child(octree.root(), 7)) == CGAL::Bbox_3(0, 0, 0, 1.1, 1.1, 1.1));
 }
 
 void test_25_nodes() {
@@ -69,49 +69,49 @@ void test_25_nodes() {
   assert(octree.bbox(octree.root()) == CGAL::Bbox_3(-1.5, -1.5, -1.5, 1.5, 1.5, 1.5));
 
   // Compare the child nodes
-  assert(octree.bbox(octree.child(octree.index(octree.root()), 0)) == CGAL::Bbox_3(-1.5, -1.5, -1.5, 0, 0, 0));
-  assert(octree.bbox(octree.child(octree.index(octree.root()), 1)) == CGAL::Bbox_3(0, -1.5, -1.5, 1.5, 0, 0));
-  assert(octree.bbox(octree.child(octree.index(octree.root()), 2)) == CGAL::Bbox_3(-1.5, 0, -1.5, 0, 1.5, 0));
-  assert(octree.bbox(octree.child(octree.index(octree.root()), 3)) == CGAL::Bbox_3(0, 0, -1.5, 1.5, 1.5, 0));
-  assert(octree.bbox(octree.child(octree.index(octree.root()), 4)) == CGAL::Bbox_3(-1.5, -1.5, 0, 0, 0, 1.5));
-  assert(octree.bbox(octree.child(octree.index(octree.root()), 5)) == CGAL::Bbox_3(0, -1.5, 0, 1.5, 0, 1.5));
-  assert(octree.bbox(octree.child(octree.index(octree.root()), 6)) == CGAL::Bbox_3(-1.5, 0, 0, 0, 1.5, 1.5));
-  assert(octree.bbox(octree.child(octree.index(octree.root()), 7)) == CGAL::Bbox_3(0, 0, 0, 1.5, 1.5, 1.5));
+  assert(octree.bbox(octree.child(octree.root(), 0)) == CGAL::Bbox_3(-1.5, -1.5, -1.5, 0, 0, 0));
+  assert(octree.bbox(octree.child(octree.root(), 1)) == CGAL::Bbox_3(0, -1.5, -1.5, 1.5, 0, 0));
+  assert(octree.bbox(octree.child(octree.root(), 2)) == CGAL::Bbox_3(-1.5, 0, -1.5, 0, 1.5, 0));
+  assert(octree.bbox(octree.child(octree.root(), 3)) == CGAL::Bbox_3(0, 0, -1.5, 1.5, 1.5, 0));
+  assert(octree.bbox(octree.child(octree.root(), 4)) == CGAL::Bbox_3(-1.5, -1.5, 0, 0, 0, 1.5));
+  assert(octree.bbox(octree.child(octree.root(), 5)) == CGAL::Bbox_3(0, -1.5, 0, 1.5, 0, 1.5));
+  assert(octree.bbox(octree.child(octree.root(), 6)) == CGAL::Bbox_3(-1.5, 0, 0, 0, 1.5, 1.5));
+  assert(octree.bbox(octree.child(octree.root(), 7)) == CGAL::Bbox_3(0, 0, 0, 1.5, 1.5, 1.5));
 
   // Compare children of the first child
-  assert(octree.bbox(octree.child(octree.child(octree.index(octree.root()), 0), 0)) ==
+  assert(octree.bbox(octree.child(octree.child(octree.root(), 0), 0)) ==
          CGAL::Bbox_3(-1.5, -1.5, -1.5, -0.75, -0.75, -0.75));
-  assert(octree.bbox(octree.child(octree.child(octree.index(octree.root()), 0), 1)) ==
+  assert(octree.bbox(octree.child(octree.child(octree.root(), 0), 1)) ==
          CGAL::Bbox_3(-0.75, -1.5, -1.5, 0, -0.75, -0.75));
-  assert(octree.bbox(octree.child(octree.child(octree.index(octree.root()), 0), 2)) ==
+  assert(octree.bbox(octree.child(octree.child(octree.root(), 0), 2)) ==
          CGAL::Bbox_3(-1.5, -0.75, -1.5, -0.75, 0, -0.75));
-  assert(octree.bbox(octree.child(octree.child(octree.index(octree.root()), 0), 3)) ==
+  assert(octree.bbox(octree.child(octree.child(octree.root(), 0), 3)) ==
          CGAL::Bbox_3(-0.75, -0.75, -1.5, 0, 0, -0.75));
-  assert(octree.bbox(octree.child(octree.child(octree.index(octree.root()), 0), 4)) ==
+  assert(octree.bbox(octree.child(octree.child(octree.root(), 0), 4)) ==
          CGAL::Bbox_3(-1.5, -1.5, -0.75, -0.75, -0.75, 0));
-  assert(octree.bbox(octree.child(octree.child(octree.index(octree.root()), 0), 5)) ==
+  assert(octree.bbox(octree.child(octree.child(octree.root(), 0), 5)) ==
          CGAL::Bbox_3(-0.75, -1.5, -0.75, 0, -0.75, 0));
-  assert(octree.bbox(octree.child(octree.child(octree.index(octree.root()), 0), 6)) ==
+  assert(octree.bbox(octree.child(octree.child(octree.root(), 0), 6)) ==
          CGAL::Bbox_3(-1.5, -0.75, -0.75, -0.75, 0, 0));
-  assert(octree.bbox(octree.child(octree.child(octree.index(octree.root()), 0), 7)) ==
+  assert(octree.bbox(octree.child(octree.child(octree.root(), 0), 7)) ==
          CGAL::Bbox_3(-0.75, -0.75, -0.75, 0, 0, 0));
 
   // Compare children of the last child
-  assert(octree.bbox(octree.child(octree.child(octree.index(octree.root()), 7), 0)) ==
+  assert(octree.bbox(octree.child(octree.child(octree.root(), 7), 0)) ==
          CGAL::Bbox_3(0, 0, 0, 0.75, 0.75, 0.75));
-  assert(octree.bbox(octree.child(octree.child(octree.index(octree.root()), 7), 1)) ==
+  assert(octree.bbox(octree.child(octree.child(octree.root(), 7), 1)) ==
          CGAL::Bbox_3(0.75, 0, 0, 1.5, 0.75, 0.75));
-  assert(octree.bbox(octree.child(octree.child(octree.index(octree.root()), 7), 2)) ==
+  assert(octree.bbox(octree.child(octree.child(octree.root(), 7), 2)) ==
          CGAL::Bbox_3(0, 0.75, 0, 0.75, 1.5, 0.75));
-  assert(octree.bbox(octree.child(octree.child(octree.index(octree.root()), 7), 3)) ==
+  assert(octree.bbox(octree.child(octree.child(octree.root(), 7), 3)) ==
          CGAL::Bbox_3(0.75, 0.75, 0, 1.5, 1.5, 0.75));
-  assert(octree.bbox(octree.child(octree.child(octree.index(octree.root()), 7), 4)) ==
+  assert(octree.bbox(octree.child(octree.child(octree.root(), 7), 4)) ==
          CGAL::Bbox_3(0, 0, 0.75, 0.75, 0.75, 1.5));
-  assert(octree.bbox(octree.child(octree.child(octree.index(octree.root()), 7), 5)) ==
+  assert(octree.bbox(octree.child(octree.child(octree.root(), 7), 5)) ==
          CGAL::Bbox_3(0.75, 0, 0.75, 1.5, 0.75, 1.5));
-  assert(octree.bbox(octree.child(octree.child(octree.index(octree.root()), 7), 6)) ==
+  assert(octree.bbox(octree.child(octree.child(octree.root(), 7), 6)) ==
          CGAL::Bbox_3(0, 0.75, 0.75, 0.75, 1.5, 1.5));
-  assert(octree.bbox(octree.child(octree.child(octree.index(octree.root()), 7), 7)) ==
+  assert(octree.bbox(octree.child(octree.child(octree.root(), 7), 7)) ==
          CGAL::Bbox_3(0.75, 0.75, 0.75, 1.5, 1.5, 1.5));
 }
 
diff --git a/Orthtree/test/Orthtree/test_octree_copy_move_constructors.cpp b/Orthtree/test/Orthtree/test_octree_copy_move_constructors.cpp
index ae6272fe149..2b4dd730e47 100644
--- a/Orthtree/test/Orthtree/test_octree_copy_move_constructors.cpp
+++ b/Orthtree/test/Orthtree/test_octree_copy_move_constructors.cpp
@@ -25,26 +25,27 @@ int main(void)
     points.insert(*(generator++));
 
   Octree base (points, points.point_map());
-  assert (base.root().is_leaf()); // base is not refined yet
+  assert (base.is_leaf(base.root())); // base is not refined yet
 
   Octree copy1 (base);
-  assert (copy1.root().is_leaf()); // copy1 is thus not refined either
+  assert (copy1.is_leaf(copy1.root())); // copy1 is thus not refined either
   assert (base == copy1); // base should be equal to copy1
 
   base.refine();
-  assert (!base.root().is_leaf()); // base is now refined
-  assert (copy1.root().is_leaf()); // copy1 should be unaffected and still unrefined
+  assert (!base.is_leaf(base.root())); // base is now refined
+  assert (copy1.is_leaf(copy1.root())); // copy1 should be unaffected and still unrefined
   assert (base != copy1); // base should be different from copy1
 
   Octree copy2 (base);
-  assert (!copy2.root().is_leaf()); // copy2 should be refined
+  assert (!copy2.is_leaf(copy2.root())); // copy2 should be refined
   assert (base == copy2); // base should be equal to copy2
 
   Octree move (std::move(base));
-  assert (!move.root().is_leaf()); // move should be refined
-  assert (base.root().is_leaf()); // base should be back to init state (unrefined)
-  assert (copy1.root().is_leaf()); // copy1 still unaffected and still unrefined
-  assert (!copy2.root().is_leaf()); // copy2 unaffected by move and still refined
+  assert (!move.is_leaf(move.root())); // move should be refined
+  // fixme: my linter isn't happy about use-after-move
+  assert (base.is_leaf(base.root())); // base should be back to init state (unrefined)
+  assert (copy1.is_leaf(copy1.root())); // copy1 still unaffected and still unrefined
+  assert (!copy2.is_leaf(copy2.root())); // copy2 unaffected by move and still refined
   assert (move == copy2); // move should be equal to copy2
 
   return EXIT_SUCCESS;
diff --git a/Orthtree/test/Orthtree/test_octree_intersecting.cpp b/Orthtree/test/Orthtree/test_octree_intersecting.cpp
index 17df0fd1fec..3470ee68b38 100644
--- a/Orthtree/test/Orthtree/test_octree_intersecting.cpp
+++ b/Orthtree/test/Orthtree/test_octree_intersecting.cpp
@@ -68,10 +68,10 @@ int main(void) {
 
     // Check the results
     assert(4 == nodes.size());
-    assert(octree.child(octree.index(octree.root()), Octree::Traits::RIGHT_TOP_BACK) == nodes[0]);
-    assert(octree.child(octree.index(octree.root()), Octree::Traits::RIGHT_BOTTOM_FRONT) == nodes[1]);
-    assert(octree.child(octree.index(octree.root()), Octree::Traits::LEFT_TOP_FRONT) == nodes[2]);
-    assert(octree.child(octree.index(octree.root()), Octree::Traits::RIGHT_TOP_FRONT) == nodes[3]);
+    assert(octree.child(octree.root(), Octree::Traits::RIGHT_TOP_BACK) == nodes[0]);
+    assert(octree.child(octree.root(), Octree::Traits::RIGHT_BOTTOM_FRONT) == nodes[1]);
+    assert(octree.child(octree.root(), Octree::Traits::LEFT_TOP_FRONT) == nodes[2]);
+    assert(octree.child(octree.root(), Octree::Traits::RIGHT_TOP_FRONT) == nodes[3]);
   }
 
   // Intersection with a ray
@@ -86,19 +86,19 @@ int main(void) {
 
     // Check the results
     assert(8 == nodes.size());
-    assert(octree.child(octree.index(octree.root()), Octree::Traits::LEFT_BOTTOM_BACK) == nodes[0]);
+    assert(octree.child(octree.root(), Octree::Traits::LEFT_BOTTOM_BACK) == nodes[0]);
     assert(
-      octree.child(octree.child(octree.index(octree.root()),
+      octree.child(octree.child(octree.root(),
                                 Octree::Traits::RIGHT_BOTTOM_BACK),
                    Octree::Traits::LEFT_TOP_FRONT)
       == nodes[1]
     );
-    assert(octree.child(octree.index(octree.root()), Octree::Traits::LEFT_TOP_BACK) == nodes[2]);
-    assert(octree.child(octree.index(octree.root()), Octree::Traits::RIGHT_TOP_BACK) == nodes[3]);
-    assert(octree.child(octree.index(octree.root()), Octree::Traits::LEFT_BOTTOM_FRONT) == nodes[4]);
-    assert(octree.child(octree.index(octree.root()), Octree::Traits::RIGHT_BOTTOM_FRONT) == nodes[5]);
-    assert(octree.child(octree.index(octree.root()), Octree::Traits::LEFT_TOP_FRONT) == nodes[6]);
-    assert(octree.child(octree.index(octree.root()), Octree::Traits::RIGHT_TOP_FRONT) == nodes[7]);
+    assert(octree.child(octree.root(), Octree::Traits::LEFT_TOP_BACK) == nodes[2]);
+    assert(octree.child(octree.root(), Octree::Traits::RIGHT_TOP_BACK) == nodes[3]);
+    assert(octree.child(octree.root(), Octree::Traits::LEFT_BOTTOM_FRONT) == nodes[4]);
+    assert(octree.child(octree.root(), Octree::Traits::RIGHT_BOTTOM_FRONT) == nodes[5]);
+    assert(octree.child(octree.root(), Octree::Traits::LEFT_TOP_FRONT) == nodes[6]);
+    assert(octree.child(octree.root(), Octree::Traits::RIGHT_TOP_FRONT) == nodes[7]);
   }
 
   return EXIT_SUCCESS;
diff --git a/Orthtree/test/Orthtree/test_octree_locate.cpp b/Orthtree/test/Orthtree/test_octree_locate.cpp
index a43593deeeb..e3214a8ab11 100644
--- a/Orthtree/test/Orthtree/test_octree_locate.cpp
+++ b/Orthtree/test/Orthtree/test_octree_locate.cpp
@@ -26,7 +26,7 @@ void test_1_point() {
   octree.refine(10, 1);
 
   // Because there's only the root node, any point should be placed in it
-  assert(octree.index(octree.root()) == octree.locate(Point(-1, -1, -1)));
+  assert(octree.root() == octree.locate(Point(-1, -1, -1)));
 
   // These points would be placed outside the root node
 //  assert(octree.root() == octree.locate({0, 0, 0}));
@@ -52,24 +52,24 @@ void test_8_points() {
   octree.refine(10, 1);
 
   // Existing points should end up in the same place
-  assert(octree.child(octree.index(octree.root()), 0) == octree.locate({-1, -1, -1}));
-  assert(octree.child(octree.index(octree.root()), 1) == octree.locate({1, -1, -1}));
-  assert(octree.child(octree.index(octree.root()), 2) == octree.locate({-1, 1, -1}));
-  assert(octree.child(octree.index(octree.root()), 3) == octree.locate({1, 1, -1}));
-  assert(octree.child(octree.index(octree.root()), 4) == octree.locate({-1, -1, 1}));
-  assert(octree.child(octree.index(octree.root()), 5) == octree.locate({1, -1, 1}));
-  assert(octree.child(octree.index(octree.root()), 6) == octree.locate({-1, 1, 1}));
-  assert(octree.child(octree.index(octree.root()), 7) == octree.locate({1, 1, 1}));
+  assert(octree.child(octree.root(), 0) == octree.locate({-1, -1, -1}));
+  assert(octree.child(octree.root(), 1) == octree.locate({1, -1, -1}));
+  assert(octree.child(octree.root(), 2) == octree.locate({-1, 1, -1}));
+  assert(octree.child(octree.root(), 3) == octree.locate({1, 1, -1}));
+  assert(octree.child(octree.root(), 4) == octree.locate({-1, -1, 1}));
+  assert(octree.child(octree.root(), 5) == octree.locate({1, -1, 1}));
+  assert(octree.child(octree.root(), 6) == octree.locate({-1, 1, 1}));
+  assert(octree.child(octree.root(), 7) == octree.locate({1, 1, 1}));
 
   // Points adjacent to the existing points should also end up in the same place
-  assert(octree.child(octree.index(octree.root()), 0) == octree.locate({-1.1, -1.1, -1.1}));
-  assert(octree.child(octree.index(octree.root()), 1) == octree.locate({1.1, -1.1, -1.1}));
-  assert(octree.child(octree.index(octree.root()), 2) == octree.locate({-1.1, 1.1, -1.1}));
-  assert(octree.child(octree.index(octree.root()), 3) == octree.locate({1.1, 1.1, -1.1}));
-  assert(octree.child(octree.index(octree.root()), 4) == octree.locate({-1.1, -1.1, 1.1}));
-  assert(octree.child(octree.index(octree.root()), 5) == octree.locate({1.1, -1.1, 1.1}));
-  assert(octree.child(octree.index(octree.root()), 6) == octree.locate({-1.1, 1.1, 1.1}));
-  assert(octree.child(octree.index(octree.root()), 7) == octree.locate({1.1, 1.1, 1.1}));
+  assert(octree.child(octree.root(), 0) == octree.locate({-1.1, -1.1, -1.1}));
+  assert(octree.child(octree.root(), 1) == octree.locate({1.1, -1.1, -1.1}));
+  assert(octree.child(octree.root(), 2) == octree.locate({-1.1, 1.1, -1.1}));
+  assert(octree.child(octree.root(), 3) == octree.locate({1.1, 1.1, -1.1}));
+  assert(octree.child(octree.root(), 4) == octree.locate({-1.1, -1.1, 1.1}));
+  assert(octree.child(octree.root(), 5) == octree.locate({1.1, -1.1, 1.1}));
+  assert(octree.child(octree.root(), 6) == octree.locate({-1.1, 1.1, 1.1}));
+  assert(octree.child(octree.root(), 7) == octree.locate({1.1, 1.1, 1.1}));
 
 }
 
@@ -93,28 +93,28 @@ void test_10_points() {
   octree.refine(10, 1);
 
   // Existing points should end up in the same place
-  assert(octree.child(octree.index(octree.root()), 0) == octree.locate({-1, -1, -1}));
-  assert(octree.child(octree.index(octree.root()), 1) == octree.locate({1, -1, -1}));
-  assert(octree.child(octree.index(octree.root()), 2) == octree.locate({-1, 1, -1}));
-  assert(octree.child(octree.child(octree.child(octree.index(octree.root()), 3), 3), 3) ==
+  assert(octree.child(octree.root(), 0) == octree.locate({-1, -1, -1}));
+  assert(octree.child(octree.root(), 1) == octree.locate({1, -1, -1}));
+  assert(octree.child(octree.root(), 2) == octree.locate({-1, 1, -1}));
+  assert(octree.child(octree.child(octree.child(octree.root(), 3), 3), 3) ==
          octree.locate({1, 1, -1}));
-  assert(octree.child(octree.child(octree.child(octree.index(octree.root()), 4), 4), 4) ==
+  assert(octree.child(octree.child(octree.child(octree.root(), 4), 4), 4) ==
          octree.locate({-1, -1, 1}));
-  assert(octree.child(octree.index(octree.root()), 5) == octree.locate({1, -1, 1}));
-  assert(octree.child(octree.index(octree.root()), 6) == octree.locate({-1, 1, 1}));
-  assert(octree.child(octree.index(octree.root()), 7) == octree.locate({1, 1, 1}));
+  assert(octree.child(octree.root(), 5) == octree.locate({1, -1, 1}));
+  assert(octree.child(octree.root(), 6) == octree.locate({-1, 1, 1}));
+  assert(octree.child(octree.root(), 7) == octree.locate({1, 1, 1}));
 
   // Points adjacent to the existing points might end up in different places
-  assert(octree.child(octree.index(octree.root()), 0) == octree.locate({-1.1, -1.1, -1.1}));
-  assert(octree.child(octree.index(octree.root()), 1) == octree.locate({1.1, -1.1, -1.1}));
-  assert(octree.child(octree.index(octree.root()), 2) == octree.locate({-1.1, 1.1, -1.1}));
-  assert(octree.child(octree.child(octree.child(octree.index(octree.root()), 3), 3), 3) ==
+  assert(octree.child(octree.root(), 0) == octree.locate({-1.1, -1.1, -1.1}));
+  assert(octree.child(octree.root(), 1) == octree.locate({1.1, -1.1, -1.1}));
+  assert(octree.child(octree.root(), 2) == octree.locate({-1.1, 1.1, -1.1}));
+  assert(octree.child(octree.child(octree.child(octree.root(), 3), 3), 3) ==
          octree.locate({1.1, 1.1, -1.1}));
-  assert(octree.child(octree.child(octree.child(octree.index(octree.root()), 4), 4), 4) ==
+  assert(octree.child(octree.child(octree.child(octree.root(), 4), 4), 4) ==
          octree.locate({-1.1, -1.1, 1.1}));
-  assert(octree.child(octree.index(octree.root()), 5) == octree.locate({1.1, -1.1, 1.1}));
-  assert(octree.child(octree.index(octree.root()), 6) == octree.locate({-1.1, 1.1, 1.1}));
-  assert(octree.child(octree.index(octree.root()), 7) == octree.locate({1.1, 1.1, 1.1}));
+  assert(octree.child(octree.root(), 5) == octree.locate({1.1, -1.1, 1.1}));
+  assert(octree.child(octree.root(), 6) == octree.locate({-1.1, 1.1, 1.1}));
+  assert(octree.child(octree.root(), 7) == octree.locate({1.1, 1.1, 1.1}));
 
 }
 
diff --git a/Orthtree/test/Orthtree/test_octree_refine.cpp b/Orthtree/test/Orthtree/test_octree_refine.cpp
index 63ca3a31fa7..543c99a5a6e 100644
--- a/Orthtree/test/Orthtree/test_octree_refine.cpp
+++ b/Orthtree/test/Orthtree/test_octree_refine.cpp
@@ -37,7 +37,7 @@ void test_1_point() {
   octree.refine(10, 1);
 
   // Check that the root node was never split
-  assert(octree.root().is_leaf());
+  assert(octree.is_leaf(octree.root()));
   assert(0 == octree.depth());
 }
 
@@ -54,7 +54,7 @@ void test_2_points() {
 
   // The octree should have been split once
   Octree other(points, points.point_map());
-  other.split(other.index(other.root()));
+  other.split(other.root());
   assert(Octree::is_topology_equal(other, octree));
   assert(1 == octree.depth());
 }
@@ -73,15 +73,15 @@ void test_4_points() {
 
   // The octree should have been split once on the first level, and twice on the second
   Octree other(points, points.point_map());
-  other.split(other.index(other.root()));
-  other.split(other.child(other.index(other.root()), 3));
-  other.split(other.child(other.index(other.root()), 7));
+  other.split(other.root());
+  other.split(other.child(other.root(), 3));
+  other.split(other.child(other.root(), 7));
   assert(Octree::is_topology_equal(other, octree));
   assert(2 == octree.depth());
 
   // Applying another splitting criterion shouldn't reset the tree.
   octree.refine(Split_nth_child_of_root(2));
-  other.split(other.child(other.index(other.root()), 2));
+  other.split(other.child(other.root(), 2));
   assert(Octree::is_topology_equal(other, octree));
 
 }
diff --git a/Orthtree/test/Orthtree/test_octree_traverse.cpp b/Orthtree/test/Orthtree/test_octree_traverse.cpp
index 82abb26b12a..f26625db46a 100644
--- a/Orthtree/test/Orthtree/test_octree_traverse.cpp
+++ b/Orthtree/test/Orthtree/test_octree_traverse.cpp
@@ -30,7 +30,7 @@ bool test_preorder_1_node() {
 
   // Check each item in the range
   auto iter = nodes.begin();
-  assert(*iter == octree.index(octree.root()));
+  assert(*iter == octree.root());
 
   return true;
 }
@@ -51,10 +51,10 @@ bool test_preorder_9_nodes() {
 
   // Check each item in the range
   auto iter = nodes.begin();
-  assert(*iter == octree.index(octree.root()));
+  assert(*iter == octree.root());
   for (int i = 0; i < 8; ++i) {
     iter++;
-    assert(*iter == octree.child(octree.index(octree.root()), i));
+    assert(*iter == octree.child(octree.root(), i));
   }
 
   return true;
@@ -77,7 +77,7 @@ bool test_level_9_nodes() {
   // Check each item in the range
   auto iter = nodes.begin();
   for (int i = 0; i < 8; ++i) {
-    assert(*iter == octree.child(octree.index(octree.root()), i));
+    assert(*iter == octree.child(octree.root(), i));
     iter++;
   }
 
@@ -102,30 +102,30 @@ bool test_preorder_25_nodes() {
 
   // Check each item in the range
   auto iter = nodes.begin();
-  assert(*iter == octree.index(octree.root()));
+  assert(*iter == octree.root());
   iter++;
-  assert(*iter == octree.child(octree.index(octree.root()), 0));
+  assert(*iter == octree.child(octree.root(), 0));
   iter++;
-  assert(*iter == octree.child(octree.index(octree.root()), 1));
+  assert(*iter == octree.child(octree.root(), 1));
   iter++;
-  assert((*iter == octree.child(octree.index(octree.root()), 2)));
+  assert((*iter == octree.child(octree.root(), 2)));
   iter++;
-  assert(*iter == octree.child(octree.index(octree.root()), 3));
+  assert(*iter == octree.child(octree.root(), 3));
   for (int i = 0; i < 8; ++i) {
     iter++;
-    assert(*iter == octree.child(octree.child(octree.index(octree.root()), 3), i));
+    assert(*iter == octree.child(octree.child(octree.root(), 3), i));
   }
   iter++;
-  assert((*iter == octree.child(octree.index(octree.root()), 4)));
+  assert((*iter == octree.child(octree.root(), 4)));
   iter++;
-  assert((*iter == octree.child(octree.index(octree.root()), 5)));
+  assert((*iter == octree.child(octree.root(), 5)));
   iter++;
-  assert((*iter == octree.child(octree.index(octree.root()), 6)));
+  assert((*iter == octree.child(octree.root(), 6)));
   iter++;
-  assert((*iter == octree.child(octree.index(octree.root()), 7)));
+  assert((*iter == octree.child(octree.root(), 7)));
   for (int i = 0; i < 8; ++i) {
     iter++;
-    assert(*iter == octree.child(octree.child(octree.index(octree.root()), 7), i));
+    assert(*iter == octree.child(octree.child(octree.root(), 7), i));
   }
 
   return true;