diff --git a/Orthtree/include/CGAL/Orthtree.h b/Orthtree/include/CGAL/Orthtree.h
index 8751c15e97e..2d00bd383a8 100644
--- a/Orthtree/include/CGAL/Orthtree.h
+++ b/Orthtree/include/CGAL/Orthtree.h
@@ -126,7 +126,7 @@ public:
   /*!
    * \brief A predicate that determines whether a node must be split when refining a tree.
    */
-  typedef std::function<bool(Node)> Split_predicate;
+  typedef std::function<bool(Node_index, const Self &)> Split_predicate;
 
   /*!
    * \brief A model of `ConstRange` whose value type is `Node`.
@@ -342,7 +342,7 @@ public:
       todo.pop();
 
       // Check if this node needs to be processed
-      if (split_predicate(m_nodes[current])) {
+      if (split_predicate(current, *this)) {
 
         // Check if we've reached a new max depth
         if (depth(current) == depth()) {
@@ -458,7 +458,6 @@ public:
 
     \return a const reference to the root node of the tree.
    */
-  // todo: return index instead of ref
   Node_index root() const { return 0; }
 
   Node_index index(const Node& node) const {
@@ -819,7 +818,7 @@ public:
     // Split the node to create children
     using Local_coordinates = typename Node::Local_coordinates;
     for (int i = 0; i < Degree::value; i++) {
-      m_nodes.emplace_back(n, global_coordinates(n), depth(n) + 1, Local_coordinates{i});
+      m_nodes.emplace_back(n, global_coordinates(n), depth(n) + 1, Local_coordinates{(unsigned long) i});
     }
     // todo: this assumes that the new nodes are always allocated at the end
     m_nodes[n].m_children_index = m_nodes.size() - Degree::value;
diff --git a/Orthtree/include/CGAL/Orthtree/Node.h b/Orthtree/include/CGAL/Orthtree/Node.h
index ca6bce35e5a..c94a1019c4c 100644
--- a/Orthtree/include/CGAL/Orthtree/Node.h
+++ b/Orthtree/include/CGAL/Orthtree/Node.h
@@ -229,7 +229,7 @@ public:
    * \param rhs node to compare with
    * \return whether the nodes have different topology.
    */
-  bool operator==(const Self& rhs) const = default;
+  bool operator==(const Self& rhs) const = default; // todo: this doesn't work in C++17
 
   friend std::ostream& operator<<(std::ostream& os, const Self& node) {
     return internal::print_orthtree_node(os, node);
diff --git a/Orthtree/include/CGAL/Orthtree/Split_predicates.h b/Orthtree/include/CGAL/Orthtree/Split_predicates.h
index e8d93415d5e..766a76e3570 100644
--- a/Orthtree/include/CGAL/Orthtree/Split_predicates.h
+++ b/Orthtree/include/CGAL/Orthtree/Split_predicates.h
@@ -45,6 +45,15 @@ public:
   bool operator()(const Node &n) const {
     return (n.size() > m_bucket_size);
   }
+
+  /*!
+    \brief returns `true` if `i` should be split, `false` otherwise.
+   */
+  template<typename Node_index, typename Tree>
+  bool operator()(Node_index i, const Tree &tree) const {
+    return (tree.points(i).size() > m_bucket_size);
+  }
+
 };
 
 /*!
@@ -71,6 +80,15 @@ public:
   bool operator()(const Node &n) const {
     return n.depth() < m_max_depth;
   }
+
+  /*!
+    \brief returns `true` if `i` should be split, `false` otherwise.
+   */
+  template<typename Node_index, typename Tree>
+  bool operator()(Node_index i, const Tree &tree) const {
+    return (tree.depth(i) < m_max_depth);
+  }
+
 };
 
 /*!
@@ -108,6 +126,17 @@ public:
     std::size_t depth = n.depth();
     return (num_points > m_bucket_size && depth < m_max_depth);
   }
+
+  /*!
+    \brief returns `true` if `i` should be split, `false` otherwise.
+   */
+  template<typename Node_index, typename Tree>
+  bool operator()(Node_index i, const Tree &tree) const {
+    std::size_t num_points = tree.points(i).size();
+    std::size_t depth = tree.depth(i);
+    return (num_points > m_bucket_size && depth < m_max_depth);
+  }
+
 };
 
 } // Orthtrees
diff --git a/Orthtree/test/Orthtree/test_octree_refine.cpp b/Orthtree/test/Orthtree/test_octree_refine.cpp
index 543c99a5a6e..2e248262a11 100644
--- a/Orthtree/test/Orthtree/test_octree_refine.cpp
+++ b/Orthtree/test/Orthtree/test_octree_refine.cpp
@@ -24,6 +24,11 @@ public:
   bool operator()(const Node& node) const {
     return (node.depth() == 1 && node.local_coordinates().to_ulong() == m_n);
   }
+
+  template<typename Node_index, typename Tree>
+  bool operator()(Node_index i, const Tree &tree) const {
+    return (tree.depth(i) == 1 && tree.local_coordinates(i).to_ulong() == m_n);
+  }
 };
 
 void test_1_point() {
diff --git a/Orthtree/test/Orthtree/test_octree_traverse.cpp b/Orthtree/test/Orthtree/test_octree_traverse.cpp
index f26625db46a..5ad71f57f53 100644
--- a/Orthtree/test/Orthtree/test_octree_traverse.cpp
+++ b/Orthtree/test/Orthtree/test_octree_traverse.cpp
@@ -72,7 +72,7 @@ bool test_level_9_nodes() {
   octree.refine(10, 1);
 
   // Create the range
-  auto nodes = octree.traverse_indices<Level_traversal>(1);
+  auto nodes = octree.traverse_indices<Level_traversal>(static_cast<std::size_t>(1));
 
   // Check each item in the range
   auto iter = nodes.begin();