diff --git a/AABB_tree/include/CGAL/AABB_tree.h b/AABB_tree/include/CGAL/AABB_tree.h
index 0fe6f6bf5de..d2ed637719f 100644
--- a/AABB_tree/include/CGAL/AABB_tree.h
+++ b/AABB_tree/include/CGAL/AABB_tree.h
@@ -137,6 +137,13 @@ namespace CGAL {
     void build(T&& ...);
 #ifndef DOXYGEN_RUNNING
     void build();
+
+    /// triggers the (re)construction of the tree similarly to a call to `build()`
+    /// but the traits functors `Compute_bbox` and `Split_primitives` are ignored
+    /// and `compute_bbox` and `split_primitives` are used instead.
+    template <class ComputeBbox, class SplitPrimitives>
+    void custom_build(const ComputeBbox& compute_bbox,
+                      const SplitPrimitives& split_primitives);
 #endif
     ///@}
 
@@ -697,9 +704,19 @@ public:
 #endif
   }
 
-  // Build the data structure, after calls to insert(..)
   template<typename Tr>
   void AABB_tree<Tr>::build()
+  {
+    custom_build(m_traits.compute_bbox_object(),
+                 m_traits.split_primitives_object());
+  }
+
+  // Build the data structure, after calls to insert(..)
+  template<typename Tr>
+  template <class ComputeBbox, class SplitPrimitives>
+  void AABB_tree<Tr>::custom_build(
+    const ComputeBbox& compute_bbox,
+    const SplitPrimitives& split_primitives)
   {
     clear_nodes();
     if(m_primitives.size() > 1) {
@@ -716,7 +733,10 @@ public:
 
       // constructs the tree
       m_p_root_node->expand(m_primitives.begin(), m_primitives.end(),
-                m_primitives.size(), m_traits);
+                            m_primitives.size(),
+                            compute_bbox,
+                            split_primitives,
+                            m_traits);
     }
 #ifdef CGAL_HAS_THREADS
     m_atomic_need_build.store(false, std::memory_order_release); // in case build() is triggered by a call to root_node()
diff --git a/AABB_tree/include/CGAL/internal/AABB_tree/AABB_node.h b/AABB_tree/include/CGAL/internal/AABB_tree/AABB_node.h
index 3844166ed50..6fcfef9ede8 100644
--- a/AABB_tree/include/CGAL/internal/AABB_tree/AABB_node.h
+++ b/AABB_tree/include/CGAL/internal/AABB_tree/AABB_node.h
@@ -56,10 +56,12 @@ public:
    *
    * [first,last[ is the range of primitives to be added to the tree.
    */
-  template<typename ConstPrimitiveIterator>
+  template<typename ConstPrimitiveIterator, typename ComputeBbox, typename SplitPrimitives>
   void expand(ConstPrimitiveIterator first,
               ConstPrimitiveIterator beyond,
               const std::size_t range,
+              const ComputeBbox& compute_bbox,
+              const SplitPrimitives& split_primitives,
               const AABBTraits&);
 
   /**
@@ -117,19 +119,20 @@ private:
 
 };  // end class AABB_node
 
-
 template<typename Tr>
-template<typename ConstPrimitiveIterator>
+template<typename ConstPrimitiveIterator, typename ComputeBbox, typename SplitPrimitives>
 void
 AABB_node<Tr>::expand(ConstPrimitiveIterator first,
                       ConstPrimitiveIterator beyond,
                       const std::size_t range,
+                      const ComputeBbox& compute_bbox,
+                      const SplitPrimitives& split_primitives,
                       const Tr& traits)
 {
-  m_bbox = traits.compute_bbox_object()(first, beyond);
+  m_bbox = compute_bbox(first, beyond);
 
   // sort primitives along longest axis aabb
-  traits.split_primitives_object()(first, beyond, m_bbox);
+  split_primitives(first, beyond, m_bbox);
 
   switch(range)
   {
@@ -140,14 +143,14 @@ AABB_node<Tr>::expand(ConstPrimitiveIterator first,
   case 3:
     m_p_left_child = &(*first);
     m_p_right_child = static_cast<Node*>(this)+1;
-    right_child().expand(first+1, beyond, 2,traits);
+    right_child().expand(first+1, beyond, 2, compute_bbox, split_primitives, traits);
     break;
   default:
     const std::size_t new_range = range/2;
     m_p_left_child = static_cast<Node*>(this) + 1;
     m_p_right_child = static_cast<Node*>(this) + new_range;
-    left_child().expand(first, first + new_range, new_range,traits);
-    right_child().expand(first + new_range, beyond, range - new_range,traits);
+    left_child().expand(first, first + new_range, new_range, compute_bbox, split_primitives, traits);
+    right_child().expand(first + new_range, beyond, range - new_range, compute_bbox, split_primitives, traits);
   }
 }