diff --git a/Orthtree/examples/Orthtree/orthtree_build.cpp b/Orthtree/examples/Orthtree/orthtree_build.cpp
index a721b2c6723..3af483eb8fc 100644
--- a/Orthtree/examples/Orthtree/orthtree_build.cpp
+++ b/Orthtree/examples/Orthtree/orthtree_build.cpp
@@ -12,7 +12,7 @@ typedef CGAL::Epick_d<Dimension> Kernel;
 typedef Kernel::Point_d Point_d;
 typedef std::vector<Point_d> Point_vector;
 
-typedef CGAL::Orthtree_traits_point_d<Kernel, Dimension, Point_vector> Traits;
+typedef CGAL::Orthtree_traits_point<Kernel, Point_vector> Traits;
 typedef CGAL::Orthtree<Traits> Orthtree;
 
 int main()
diff --git a/Orthtree/examples/Orthtree/quadtree_build_manually.cpp b/Orthtree/examples/Orthtree/quadtree_build_manually.cpp
index 55be9e46ed6..f0fe27d1f20 100644
--- a/Orthtree/examples/Orthtree/quadtree_build_manually.cpp
+++ b/Orthtree/examples/Orthtree/quadtree_build_manually.cpp
@@ -4,7 +4,7 @@
 #include <CGAL/Dimension.h>
 
 #include <CGAL/Orthtree.h>
-#include <CGAL/Orthtree_traits_2_base.h>
+#include <CGAL/Orthtree_traits_base_for_dimension.h>
 
 using Kernel = CGAL::Simple_cartesian<double>;
 
@@ -13,15 +13,15 @@ namespace CGAL {
 struct empty_type {
 };
 
-template <typename K>
-struct Orthtree_traits_empty_2 : public Orthtree_traits_2_base<K> {
+template <typename K, typename DimensionTag>
+struct Orthtree_traits_empty : public Orthtree_traits_base_for_dimension<K, DimensionTag> {
 
-  using Self = Orthtree_traits_empty_2<K>;
+  using Self = Orthtree_traits_empty<K, DimensionTag>;
   using Tree = Orthtree<Self>;
 
   using Node_data = std::array<empty_type, 0>;
 
-  Orthtree_traits_empty_2(typename Self::Bbox_d bbox) : m_bbox(bbox) {};
+  Orthtree_traits_empty(typename Self::Bbox_d bbox) : m_bbox(bbox) {};
 
   auto root_node_bbox_object() const {
     return [&]() -> typename Self::Bbox_d { return m_bbox; };
@@ -40,7 +40,7 @@ private:
 };
 }
 
-using EmptyQuadtree = CGAL::Orthtree<CGAL::Orthtree_traits_empty_2<Kernel>>;
+using EmptyQuadtree = CGAL::Orthtree<CGAL::Orthtree_traits_empty<Kernel, CGAL::Dimension_tag<2>>>;
 
 int main() {
 
diff --git a/Orthtree/include/CGAL/Octree.h b/Orthtree/include/CGAL/Octree.h
index 9911668d67b..324d655e1d9 100644
--- a/Orthtree/include/CGAL/Octree.h
+++ b/Orthtree/include/CGAL/Octree.h
@@ -43,7 +43,7 @@ template <
 #ifdef DOXYGEN_RUNNING
   class Octree;
 #else
-using Octree = Orthtree<Orthtree_traits_point_3<GeomTraits, PointRange, PointMap>>;
+using Octree = Orthtree<Orthtree_traits_point<GeomTraits, PointRange, PointMap, Dimension_tag<3>>>;
 #endif
 
 } // namespace CGAL
diff --git a/Orthtree/include/CGAL/Orthtree_traits_base_for_dimension.h b/Orthtree/include/CGAL/Orthtree_traits_base_for_dimension.h
new file mode 100644
index 00000000000..fb7d4f6b2f9
--- /dev/null
+++ b/Orthtree/include/CGAL/Orthtree_traits_base_for_dimension.h
@@ -0,0 +1,196 @@
+// Copyright (c) 2023  INRIA (France).
+// All rights reserved.
+//
+// This file is part of CGAL (www.cgal.org).
+//
+// $URL$
+// $Id$
+// SPDX-License-Identifier: GPL-3.0-or-later OR LicenseRef-Commercial
+//
+// Author(s)     : Jackson Campolattaro
+
+#ifndef ORTHTREE_EXAMPLES_ORTHTREE_TRAITS_BASE_FOR_DIMENSION_H
+#define ORTHTREE_EXAMPLES_ORTHTREE_TRAITS_BASE_FOR_DIMENSION_H
+
+#include <CGAL/license/Orthtree.h>
+
+#include <CGAL/Dimension.h>
+#include <CGAL/Bbox_2.h>
+#include <CGAL/Point_set_2.h>
+#include <CGAL/Orthtree/Cartesian_ranges.h>
+
+namespace CGAL {
+
+template <typename K, typename DimensionTag>
+struct Orthtree_traits_base_for_dimension;
+
+template <typename K, typename DimensionTag>
+struct Orthtree_traits_base_for_dimension {
+  /// \name Types
+  /// @{
+  using Dimension = DimensionTag;
+  using FT = typename K::FT;
+  using Point_d = typename K::Point_d;
+  using Bbox_d = typename K::Iso_box_d;
+  using Sphere_d = typename K::Sphere_d;
+  using Cartesian_const_iterator_d = typename K::Cartesian_const_iterator_d;
+  /*!
+    Adjacency type.
+
+    \note This type is used to identify adjacency directions with
+    easily understandable keywords (left, right, up, etc.) and is thus
+    mainly useful for `Orthtree_traits_2` and `Orthtree_traits_3`. In
+    higher dimensions, such keywords do not exist and this type is
+    simply an integer. Conversions from this integer to bitsets still
+    work but do not provide any easier API for adjacency selection.
+  */
+  using Adjacency = int;
+  /// @}
+
+  /// \name Operations
+  /// @{
+  auto construct_point_d_object() const {
+    return [](auto... Args) -> Point_d {
+      std::initializer_list<FT> args_list{Args...};
+      return Point_d{args_list.size(), args_list.begin(), args_list.end()};
+    };
+  }
+  /// @}
+};
+
+template <typename K>
+struct Orthtree_traits_base_for_dimension<K, Dimension_tag<2>> {
+  /// \name Types
+  /// @{
+  using Dimension = Dimension_tag<2>;
+  using FT = typename K::FT;
+  using Point_d = typename K::Point_2;
+  using Bbox_d = typename K::Iso_rectangle_2;
+  using Sphere_d = typename K::Circle_2;
+  using Cartesian_const_iterator_d = typename K::Cartesian_const_iterator_2;
+  /*!
+   * \brief Two directions along each axis in Cartesian space, relative to a node.
+   *
+   * Directions are mapped to numbers as 2-bit integers.
+   *
+   * The first bit indicates the axis (0 = x, 1 = y),
+   * the second bit indicates the direction along that axis (0 = -, 1 = +).
+   *
+   * The following diagram may be a useful reference:
+   *
+   *            3 *
+   *              |
+   *              |                    y+
+   *              |                    *
+   *     0 *------+------* 1           |
+   *              |                    |
+   *              |                    +-----* x+
+   *              |
+   *              * 2
+   *
+   * This lookup table may also be helpful:
+   *
+   * | Direction | bitset | number | Enum  |
+   * | --------- | ------ | ------ | ----- |
+   * | `-x`      |  00    | 0      | LEFT  |
+   * | `+x`      |  01    | 1      | RIGHT |
+   * | `-y`      |  10    | 2      | DOWN  |
+   * | `+y`      |  11    | 3      | UP    |
+   */
+  enum Adjacency {
+    LEFT,
+    RIGHT,
+    DOWN,
+    UP
+  };
+  /// @}
+
+  /// \name Operations
+  /// @{
+  auto construct_point_d_object() const {
+    return [](const FT& x, const FT& y) -> Point_d {
+      return {x, y};
+    };
+  }
+  /// @}
+};
+
+template <typename K>
+struct Orthtree_traits_base_for_dimension<K, Dimension_tag<3>> {
+  /// \name Types
+  /// @{
+  using GeomTraits = K;
+  using Dimension = Dimension_tag<3>;
+  using FT = typename K::FT;
+  using Point_d = typename K::Point_3;
+  using Bbox_d = typename K::Iso_cuboid_3;
+  using Sphere_d = typename K::Sphere_3;
+  using Cartesian_const_iterator_d = typename K::Cartesian_const_iterator_3;
+  /*!
+   * \brief Two directions along each axis in Cartesian space, relative to a node.
+   *
+   * Directions are mapped to numbers as 3-bit integers,
+   * though the numbers 6 and 7 are not used because there are only 6 different directions.
+   *
+   * The first two bits indicate the axis (00 = x, 01 = y, 10 = z),
+   * the third bit indicates the direction along that axis (0 = -, 1 = +).
+   *
+   * The following diagram may be a useful reference:
+   *
+   *            3 *
+   *              |  * 5
+   *              | /                  y+
+   *              |/                   *  z+
+   *     0 *------+------* 1           | *
+   *             /|                    |/
+   *            / |                    +-----* x+
+   *         4 *  |
+   *              * 2
+   *
+   * This lookup table may also be helpful:
+   *
+   * | Direction | bitset | number | Enum  |
+   * | --------- | ------ | ------ | ----- |
+   * | `-x`      | 000    | 0      | LEFT  |
+   * | `+x`      | 001    | 1      | RIGHT |
+   * | `-y`      | 010    | 2      | DOWN  |
+   * | `+y`      | 011    | 3      | UP    |
+   * | `-z`      | 100    | 4      | BACK  |
+   * | `+z`      | 101    | 5      | FRONT |
+   */
+  enum Adjacency {
+    LEFT,
+    RIGHT,
+    DOWN,
+    UP,
+    BACK,
+    FRONT
+  };
+  /// \cond SKIP_IN_MANUAL
+  enum Child {
+    LEFT_BOTTOM_BACK,
+    RIGHT_BOTTOM_BACK,
+    LEFT_TOP_BACK,
+    RIGHT_TOP_BACK,
+    LEFT_BOTTOM_FRONT,
+    RIGHT_BOTTOM_FRONT,
+    LEFT_TOP_FRONT,
+    RIGHT_TOP_FRONT
+  };
+  /// \endcond
+  /// @}
+
+  /// \name Operations
+  /// @{
+  auto construct_point_d_object() const {
+    return [](const FT& x, const FT& y, const FT& z) -> Point_d {
+      return {x, y, z};
+    };
+  }
+  /// @}
+};
+
+
+}
+
+#endif //ORTHTREE_EXAMPLES_ORTHTREE_TRAITS_BASE_FOR_DIMENSION_H
diff --git a/Orthtree/include/CGAL/Orthtree_traits_face_graph.h b/Orthtree/include/CGAL/Orthtree_traits_face_graph.h
index 084baed7b4d..144564ebf79 100644
--- a/Orthtree/include/CGAL/Orthtree_traits_face_graph.h
+++ b/Orthtree/include/CGAL/Orthtree_traits_face_graph.h
@@ -15,7 +15,7 @@
 
 #include <CGAL/license/Orthtree.h>
 
-#include <CGAL/Orthtree_traits_3_base.h>
+#include <CGAL/Orthtree_traits_base_for_dimension.h>
 
 #include <CGAL/Dimension.h>
 #include <CGAL/Bbox_3.h>
@@ -23,8 +23,15 @@
 namespace CGAL {
 
 template <class PolygonMesh, class VPM>
-struct Orthtree_traits_face_graph
-  : public Orthtree_traits_3_base<typename Kernel_traits<typename boost::property_traits<VPM>::value_type>::type> {
+struct Orthtree_traits_face_graph : public Orthtree_traits_base_for_dimension<
+  typename Kernel_traits<typename boost::property_traits<VPM>::value_type>::type,
+  Dimension_tag<3>
+  // todo: it should be possible to determine the ambient dimension automatically, but this isn't working
+//  Ambient_dimension<
+//    typename boost::property_traits<VPM>::value_type,
+//    typename Kernel_traits<typename boost::property_traits<VPM>::value_type>::type
+//  >
+> {
 
   Orthtree_traits_face_graph(const PolygonMesh& pm, VPM vpm)
     : m_pm(pm), m_vpm(vpm) {}
@@ -110,8 +117,9 @@ struct Orthtree_traits_face_graph
       if (tree.data(ni).empty()) return false;
 
       Bbox_d bb = tree.bbox(ni);
-      //TODO: we should get better version to get guarantees
+      // TODO: we should get better version to get guarantees
       // TODO: as long as the bbox is cubic you can use depth and initial size to conclude.
+      // todo (jackson): bbox is _not_ guaranteed to be cubic now, this may break in some very niche cases
       for (int i = 0; i < 3; ++i)
         if ((bb.max()[i] - bb.min()[i]) < 2 * m_min_extent)
           return false;
diff --git a/Orthtree/include/CGAL/Orthtree_traits_point.h b/Orthtree/include/CGAL/Orthtree_traits_point.h
index b94acca60b9..dd4803e07bf 100644
--- a/Orthtree/include/CGAL/Orthtree_traits_point.h
+++ b/Orthtree/include/CGAL/Orthtree_traits_point.h
@@ -19,9 +19,7 @@
 #include <CGAL/Point_set_2.h>
 #include <CGAL/Orthtree/Cartesian_ranges.h>
 
-#include <CGAL/Orthtree_traits_2_base.h>
-#include <CGAL/Orthtree_traits_3_base.h>
-#include <CGAL/Orthtree_traits_d_base.h>
+#include <CGAL/Orthtree_traits_base_for_dimension.h>
 
 namespace CGAL {
 
@@ -78,16 +76,19 @@ void reassign_points(
 template <
   typename GeomTraits,
   typename PointSet,
-  typename PointMap,
-  typename OrthtreeTraitsDimensionBase
+  typename PointMap = Identity_property_map<typename std::iterator_traits<typename PointSet::iterator>::value_type>,
+  typename DimensionTag = Ambient_dimension<
+    typename std::iterator_traits<typename PointSet::iterator>::value_type,
+    GeomTraits
+  >
 >
-struct Orthtree_traits_point : public OrthtreeTraitsDimensionBase {
+struct Orthtree_traits_point : public Orthtree_traits_base_for_dimension<GeomTraits, DimensionTag> {
 public:
 
   /// \name Types
   /// @{
 
-  using Self = Orthtree_traits_point<GeomTraits, PointSet, PointMap, OrthtreeTraitsDimensionBase>;
+  using Self = Orthtree_traits_point<GeomTraits, PointSet, PointMap, DimensionTag>;
   using Tree = Orthtree<Self>;
 
   using Node_data = boost::iterator_range<typename PointSet::iterator>;
@@ -163,31 +164,6 @@ private:
 
 };
 
-template <
-  typename GeomTraits,
-  typename PointSet,
-  typename PointMap = Identity_property_map<typename GeomTraits::Point_2>
->
-using Orthtree_traits_point_2 =
-  Orthtree_traits_point<GeomTraits, PointSet, PointMap, Orthtree_traits_2_base<GeomTraits>>;
-
-template <
-  typename GeomTraits,
-  typename PointSet,
-  typename PointMap = Identity_property_map<typename GeomTraits::Point_3>
->
-using Orthtree_traits_point_3 =
-  Orthtree_traits_point<GeomTraits, PointSet, PointMap, Orthtree_traits_3_base<GeomTraits>>;
-
-template <
-  typename GeomTraits,
-  typename DimensionTag,
-  typename PointSet,
-  typename PointMap = Identity_property_map<typename GeomTraits::Point_d>
->
-using Orthtree_traits_point_d =
-  Orthtree_traits_point<GeomTraits, PointSet, PointMap, Orthtree_traits_d_base<GeomTraits, DimensionTag>>;
-
 }
 
 
diff --git a/Orthtree/include/CGAL/Quadtree.h b/Orthtree/include/CGAL/Quadtree.h
index 52184807e3b..5ed4dc1baab 100644
--- a/Orthtree/include/CGAL/Quadtree.h
+++ b/Orthtree/include/CGAL/Quadtree.h
@@ -41,7 +41,7 @@ template <typename GeomTraits, typename PointRange,
 #ifdef DOXYGEN_RUNNING
 class Quadtree;
 #else
-using Quadtree = Orthtree<Orthtree_traits_point_2<GeomTraits, PointRange, PointMap>>;
+using Quadtree = Orthtree<Orthtree_traits_point<GeomTraits, PointRange, PointMap, Dimension_tag<2>>>;
 #endif
 
 } // namespace CGAL