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aka_tree.hh
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			/**
			* @file aka_tree.hh
			*
			* @author Alejandro M. Aragón <alejandro.aragon@epfl.ch>
			*
			* @date creation: Fri Jan 04 2013
			* @date last modification: Tue May 13 2014
			*
			* @brief bounding volume hierarchies
			*
			* @section LICENSE
			*
			* Copyright (©) 2014 EPFL (Ecole Polytechnique Fédérale de Lausanne)
			* Laboratory (LSMS - Laboratoire de Simulation en Mécanique des Solides)
			*
			* Akantu is free software: you can redistribute it and/or modify it under the
			* terms of the GNU Lesser General Public License as published by the Free
			* Software Foundation, either version 3 of the License, or (at your option) any
			* later version.
			*
			* Akantu is distributed in the hope that it will be useful, but WITHOUT ANY
			* WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR
			* A PARTICULAR PURPOSE. See the GNU Lesser General Public License for more
			* details.
			*
			* You should have received a copy of the GNU Lesser General Public License
			* along with Akantu. If not, see <http://www.gnu.org/licenses/>.
			*
			*/

			/* -------------------------------------------------------------------------- */

			#ifndef __AKANTU_TREE_HH__
			#define __AKANTU_TREE_HH__

			#include <cassert>
			#include <queue>
			#include <set>
			#include <iomanip>
			//#include <limits>

			#include "aka_ball.hh"
			#include "aka_bounding_box.hh"
			#include "aka_array.hh"

			//#define DEBUG_TREE 1


			__BEGIN_AKANTU__


			using std::cout;
			using std::endl;


			struct Tree_node_base
			{
			typedef Tree_node_base* base_ptr;
			typedef const Tree_node_base* const_base_ptr;

			base_ptr parent_;
			base_ptr left_;
			base_ptr right_;

			static base_ptr _minimum(base_ptr x) {
			while (x->left_ != 0) x = x->left_;
			return x;
			}

			static const_base_ptr _minimum(const_base_ptr x) {
			while (x->left_ != 0) x = x->left_;
			return x;
			}

			static base_ptr _maximum(base_ptr x) {
			while (x->right_ != 0) x = x->right_;
			return x;
			}

			static const_base_ptr _maximum(const_base_ptr x) {
			while (x->right_ != 0) x = x->right_;
			return x;
			}
			};

			template<typename T>
			struct Tree_node : public Tree_node_base
			{
			typedef T value_tpe;
			typedef Tree_node<value_tpe>* link_type;

			value_tpe value_;

			template<typename... Args>
			Tree_node(Args&&... args)
			: Tree_node_base(), value_(std::forward<Args>(args)...) { }
			};


			// iterators

			static Tree_node_base* local_leaf_increment(Tree_node_base* x) throw () {

			Tree_node_base* y = x->parent_;
			if (y->parent_ == x)
			return y;
			while (x == y->right_) {
			x = y;
			y = y->parent_;
			}
			if (y->parent_ == x)
			return y;
			if (x->right_ != y)
			x = y;
			if (x->right_ != 0) {
			x = x->right_;
			while (x->left_ != 0)
			x = x->left_;
			}
			return x;
			}

			template <class tree_pointer>
			tree_pointer* _tree_leaf_increment(tree_pointer* x) throw ()
			{ return local_leaf_increment(x); }

			template <class tree_pointer>
			const tree_pointer* _tree_leaf_increment(const tree_pointer* x) throw ()
			{ return local_leaf_increment(const_cast<tree_pointer*>(x)); }


			template<typename _Tp>
			struct Tree_leaf_iterator
			{
			typedef _Tp value_type;
			typedef _Tp& reference;
			typedef _Tp* pointer;

			typedef std::bidirectional_iterator_tag iterator_category;
			typedef std::ptrdiff_t difference_type;

			typedef Tree_leaf_iterator<_Tp> self_type;
			typedef Tree_node_base::base_ptr base_ptr;
			typedef Tree_node<_Tp>* link_type;

			Tree_leaf_iterator() : node_() { }

			explicit Tree_leaf_iterator(link_type x) : node_(x) { }

			reference operator*() const
			{ return static_cast<link_type>(node_)->value_; }

			pointer operator->() const
			{ return std::__addressof(static_cast<link_type>(node_)->value_); }

			self_type& operator++() {
			node_ = _tree_leaf_increment(node_);
			return *this;
			}

			self_type operator++(int) {
			self_type tmp = *this;
			node_ = _tree_leaf_increment(node_);
			return tmp;
			}

			bool operator==(const self_type& x) const
			{ return node_ == x.node_; }

			bool operator!=(const self_type& x) const
			{ return node_ != x.node_; }

			bool operator<(const self_type& x) const
			{ return node_ < x.node_; }

			base_ptr node_;
			};


			template<typename _Tp>
			struct Tree_leaf_const_iterator
			{
			typedef _Tp value_type;
			typedef const _Tp& reference;
			typedef const _Tp* pointer;

			typedef Tree_leaf_iterator<_Tp> iterator;

			typedef std::bidirectional_iterator_tag iterator_category;
			typedef std::ptrdiff_t difference_type;

			typedef Tree_leaf_const_iterator<_Tp> self_type;
			typedef Tree_node_base::const_base_ptr base_ptr;
			typedef const Tree_node<_Tp>* link_type;

			Tree_leaf_const_iterator() : node_() { }

			explicit Tree_leaf_const_iterator(link_type x) : node_(x) { }

			Tree_leaf_const_iterator(const iterator& it) : node_(it.node_) { }

			iterator _const_cast() const
			{ return iterator(static_cast<typename iterator::link_type>(const_cast<typename iterator::base_ptr>(node_))); }

			reference operator*() const
			{ return static_cast<link_type>(node_)->value_; }

			pointer operator->() const
			{ return std::__addressof(static_cast<link_type>
			(node_)->value_); }

			self_type& operator++() {
			node_ = _tree_leaf_increment(node_);
			return *this;
			}

			self_type operator++(int) {
			self_type tmp = *this;
			node_ = _tree_leaf_increment(node_);
			return tmp;
			}

			bool operator==(const self_type& x) const
			{ return node_ == x.node_; }

			bool operator!=(const self_type& x) const
			{ return node_ != x.node_; }

			bool operator<(const self_type& x) const
			{ return node_ < x.node_; }

			base_ptr node_;
			};


			template<typename _Val>
			inline bool operator==(const Tree_leaf_iterator<_Val>& x,
			const Tree_leaf_const_iterator<_Val>& y)
			{ return x.node_ == y.node_; }

			template<typename _Val>
			inline bool operator!=(const Tree_leaf_iterator<_Val>& x,
			const Tree_leaf_const_iterator<_Val>& y)
			{ return x.node_ != y.node_; }



			template <class tree_pointer>
			static tree_pointer*
			local_tree_increment(tree_pointer* x) throw ()
			{
			if (x->right_ != 0)
			{
			x = x->right_;
			while (x->left_ != 0)
			x = x->left_;
			}
			else
			{
			tree_pointer* y = x->parent_;
			while (x == y->right_)
			{
			x = y;
			y = y->parent_;
			}
			if (x->right_ != y)
			x = y;
			}
			return x;
			}


			template <class tree_pointer>
			tree_pointer* _tree_increment(tree_pointer* x) throw ()
			{ return local_tree_increment(x); }

			template <class tree_pointer>
			const tree_pointer*
			_tree_increment(const tree_pointer* x) throw ()
			{ return local_tree_increment(const_cast<tree_pointer*>(x)); }

			template <class tree_pointer>
			static tree_pointer*
			local_tree_decrement(tree_pointer* x) throw () {
			if (x->parent_->parent_ == x)
			x = x->right_;
			else if (x->left_ != 0) {
			Tree_node_base* y = x->left_;
			while (y->right_ != 0)
			y = y->right_;
			x = y;
			}
			else {
			tree_pointer* y = x->parent_;
			while (x == y->left_) {
			x = y;
			y = y->parent_;
			}
			x = y;
			}
			return x;
			}

			template <class tree_pointer>
			tree_pointer* _tree_decrement(tree_pointer* x) throw ()
			{ return local_tree_decrement(x); }

			template <class tree_pointer>
			const tree_pointer* _tree_decrement(const tree_pointer* x) throw ()
			{ return local_tree_decrement(const_cast<tree_pointer*>(x)); }


			// iterators

			template<typename _Tp>
			struct Tree_iterator
			{
			typedef _Tp value_type;
			typedef _Tp& reference;
			typedef _Tp* pointer;

			typedef std::bidirectional_iterator_tag iterator_category;
			typedef std::ptrdiff_t difference_type;

			typedef Tree_iterator<_Tp> self_type;
			typedef Tree_node_base::base_ptr base_ptr;
			typedef Tree_node<_Tp>* link_type;

			typedef Tree_leaf_iterator<value_type> leaf_iterator;


			Tree_iterator() : node_() { }

			explicit Tree_iterator(link_type x) : node_(x) { }

			Tree_iterator(const leaf_iterator& it) : node_(it.node_) { }

			reference operator*() const
			{ return static_cast<link_type>(node_)->value_; }

			pointer operator->() const
			{ return std::__addressof(static_cast<link_type>(node_)->value_); }

			self_type& operator++() {
			node_ = _tree_increment(node_);
			return *this;
			}

			self_type operator++(int) {
			self_type tmp = *this;
			node_ = _tree_increment(node_);
			return tmp;
			}

			self_type& operator--() {
			node_ = _tree_decrement(node_);
			return *this;
			}

			self_type operator--(int) {
			self_type tmp = *this;
			node_ = _tree_decrement(node_);
			return tmp;
			}


			self_type left()
			{ return self_type(static_cast<link_type>(node_->left_)); }

			self_type right()
			{ return self_type(static_cast<link_type>(node_->right_)); }

			self_type parent()
			{ return self_type(static_cast<link_type>(node_->parent_)); }

			bool is_leaf() {
			assert(node_ != nullptr);
			return node_->left_ == nullptr && node_->right_ == nullptr;
			}


			bool operator==(const self_type& x) const
			{ return node_ == x.node_; }

			bool operator!=(const self_type& x) const
			{ return node_ != x.node_; }

			bool operator<(const self_type& x) const
			{ return node_ < x.node_; }

			base_ptr node_;
			};

			template<typename _Tp>
			struct Tree_const_iterator
			{
			typedef _Tp value_type;
			typedef const _Tp& reference;
			typedef const _Tp* pointer;

			typedef Tree_iterator<_Tp> iterator;

			typedef std::bidirectional_iterator_tag iterator_category;
			typedef std::ptrdiff_t difference_type;

			typedef Tree_const_iterator<_Tp> self_type;
			typedef Tree_node_base::const_base_ptr base_ptr;
			typedef const Tree_node<_Tp>* link_type;

			Tree_const_iterator() : node_() { }

			explicit Tree_const_iterator(link_type x) : node_(x) { }

			Tree_const_iterator(const iterator& it) : node_(it.node_) { }

			iterator _const_cast() const
			{ return iterator(static_cast<typename iterator::link_type>(const_cast<typename iterator::base_ptr>(node_))); }

			reference operator*() const
			{ return static_cast<link_type>(node_)->value_; }

			pointer operator->() const
			{ return std::__addressof(static_cast<link_type>
			(node_)->value_); }

			self_type& operator++() {
			node_ = _tree_increment(node_);
			return *this;
			}

			self_type operator++(int) {
			self_type tmp = *this;
			node_ = _tree_increment(node_);
			return tmp;
			}

			self_type& operator--() {
			node_ = _tree_decrement(node_);
			return *this;
			}

			self_type operator--(int) {
			self_type tmp = *this;
			node_ = _tree_decrement(node_);
			return tmp;
			}

			self_type left()
			{ return self_type(static_cast<link_type>(node_->left_)); }

			self_type right()
			{ return self_type(static_cast<link_type>(node_->right_)); }

			self_type parent()
			{ return self_type(static_cast<link_type>(node_->parent_)); }

			bool is_leaf() {
			assert(node_ != nullptr);
			return node_->left_ == nullptr && node_->right_ == nullptr;
			}


			bool operator==(const self_type& x) const
			{ return node_ == x.node_; }

			bool operator!=(const self_type& x) const
			{ return node_ != x.node_; }

			bool operator<(const self_type& x) const
			{ return node_ < x.node_; }

			base_ptr node_;
			};

			template<typename _Val>
			inline bool operator==(const Tree_iterator<_Val>& x,
			const Tree_const_iterator<_Val>& y)
			{ return x.node_ == y.node_; }

			template<typename _Val>
			inline bool operator!=(const Tree_iterator<_Val>& x,
			const Tree_const_iterator<_Val>& y)
			{ return x.node_ != y.node_; }




			// traversal
			template <class iterator_type, class functor_type>
			void inorder(iterator_type it, functor_type& fn) {
			if (it.node_ == nullptr)
			return;
			inorder(it.left(), fn);
			fn(it);
			inorder(it.right(), fn);
			}

			template <class iterator_type, class functor_type>
			void preorder(iterator_type it, functor_type& fn) {
			if (it.node_ == nullptr)
			return;
			fn(it);
			preorder(it.left(), fn);
			preorder(it.right(), fn);
			}

			template <class iterator_type, class functor_type>
			void postorder(iterator_type it, functor_type& fn) {
			if (it.node_ == nullptr)
			return;
			postorder(it.left(), fn);
			postorder(it.right(), fn);
			fn(it);
			}



			//! Tree class
			template<typename T, template <class> class _Cost, typename _Alloc = std::allocator<T> >
			class Tree
			{
			typedef typename _Alloc::template rebind<Tree_node<T> >::other
			_Node_allocator;

			protected:
			typedef Tree_node_base* base_ptr;
			typedef const Tree_node_base* const_base_ptr;

			public:

			typedef T value_type;
			typedef value_type* pointer;
			typedef const value_type* const_pointer;
			typedef value_type& reference;
			typedef const value_type& const_reference;
			typedef Tree_node<T>* link_type;
			typedef const Tree_node<T>* const_link_type;
			typedef size_t size_type;
			typedef std::ptrdiff_t difference_type;
			typedef _Cost<value_type> cost_functor;
			typedef _Alloc allocator_type;

			_Node_allocator& _get_Node_allocator()
			{ return static_cast<_Node_allocator>(&this->impl_); }

			const _Node_allocator& _get_Node_allocator() const
			{ return static_cast<const _Node_allocator>(&this->impl_); }

			allocator_type
			get_allocator() const
			{ return allocator_type(_get_Node_allocator()); }

			private:

			link_type _get_node()
			{ return impl_._Node_allocator::allocate(1); }

			void _put_node(link_type p)
			{ impl_._Node_allocator::deallocate(p, 1); }

			void _destroy_node(link_type p) {
			_get_Node_allocator().destroy(p);
			_put_node(p);
			}

			link_type _clone_node(const_link_type x) {
			link_type tmp = _create_node(x->value_);
			tmp->left_ = 0;
			tmp->right_ = 0;
			return tmp;
			}

			template<typename... _Args>
			link_type _create_node(_Args&&... args) {

			link_type tmp = _get_node();
			try {
			_get_Node_allocator().construct(tmp, std::forward<_Args>(args)...);
			tmp->left_ = 0;
			tmp->right_ = 0;
			tmp->parent_ = 0;
			}
			catch(...) {
			_put_node(tmp);
			throw;
			}
			return tmp;
			}

			private:

			template<typename _Cost_functor>
			struct Tree_impl : public _Node_allocator
			{
			_Cost_functor cost_;
			Tree_node_base header_;
			size_type node_count_; // Keeps track of size of tree.

			Tree_impl()
			: _Node_allocator(), cost_(), header_(),
			node_count_(0)
			{ _M_initialize(); }

			Tree_impl(const _Cost_functor& comp, const _Node_allocator& a)
			: _Node_allocator(a), cost_(comp), header_(),
			node_count_(0)
			{ _M_initialize(); }

			Tree_impl(const _Cost_functor& comp, _Node_allocator&& a)
			: _Node_allocator(std::move(a)), cost_(comp),
			header_(), node_count_(0)
			{ _M_initialize(); }

			private:
			void
			_M_initialize()
			{
			this->header_.parent_ = 0;
			this->header_.left_ = &this->header_;
			this->header_.right_ = &this->header_;
			}
			};

			private:

			Tree_impl<cost_functor> impl_; //!< Tree implementation instance

			base_ptr& _root()
			{ return this->impl_.header_.parent_; }

			const_base_ptr _root() const
			{ return this->impl_.header_.parent_; }

			base_ptr& _leftmost()
			{ return this->impl_.header_.left_; }

			const_base_ptr _leftmost() const
			{ return this->impl_.header_.left_; }

			base_ptr& _rightmost()
			{ return this->impl_.header_.right_; }

			const_base_ptr _rightmost() const
			{ return this->impl_.header_.right_; }

			link_type _begin()
			{ return static_cast<link_type>(this->impl_.header_.parent_); }

			const_link_type _begin() const
			{ return static_cast<const_link_type>(this->impl_.header_.parent_); }

			link_type _end()
			{ return static_cast<link_type>(&this->impl_.header_); }

			const_link_type _end() const
			{ return static_cast<const_link_type>(&this->impl_.header_); }

			static reference _value(link_type x)
			{ return x->value_; }

			static const_reference _value(const_link_type x)
			{ return x->value_; }

			static reference _value(base_ptr x)
			{ return static_cast<link_type>(x)->value_; }

			static const_reference _value(const_base_ptr x)
			{ return static_cast<const_link_type>(x)->value_; }

			static link_type _left(base_ptr x)
			{ return static_cast<link_type>(x->left_); }

			static const_link_type _left(const_base_ptr x)
			{ return static_cast<const_link_type>(x->left_); }

			static link_type _right(base_ptr x)
			{ return static_cast<link_type>(x->right_); }

			static const_link_type _right(const_base_ptr x)
			{ return static_cast<const_link_type>(x->right_); }

			static link_type _parent(base_ptr x)
			{ return static_cast<link_type>(x->parent_); }

			static const_link_type _parent(const_base_ptr x)
			{ return static_cast<const_link_type>(x->parent_); }


			void _erase(link_type x) {
			while (x != 0)
			{
			#ifdef DEBUG_TREE
			cout<<"pointer x -> "<<x<<endl;
			cout<<"calling erase on right -> "<<_right(x)<<endl;
			cout<<"destroying node -> "<<_value(x)<<endl;
			cout<<"setting x to left -> "<<_left(x)<<endl;
			#endif
			_erase(_right(x));
			link_type y = _left(x);
			_destroy_node(x);
			--impl_.node_count_; // decrement nodes
			x = y;
			}
			}

			link_type _copy(const_link_type x, link_type p) {

			link_type top = _clone_node(x);
			top->parent_ = p;

			try
			{
			if (x->right_)
			top->right_ = _copy(_right(x), top);
			p = top;
			x = _left(x);

			while (x != 0)
			{
			link_type y = _clone_node(x);
			p->left_ = y;
			y->parent_ = p;
			if (x->right_)
			y->right_ = _copy(_right(x), y);
			p = y;
			x = _left(x);
			}
			}
			catch(...)
			{
			_erase(top);
			throw;
			}
			return top;
			}

			void _check_integrity(link_type x) {

			while (x != 0) {

			link_type r = _right(x);
			if (r)
			if (r->parent_ != x) {
			cout<<"* ERROR * Integrity check failed"<<endl;
			cout<<"Right child parent mismatch"<<endl;
			exit(1);
			}
			link_type l = _left(x);
			if (l)
			if (l->parent_ != x) {
			cout<<"* ERROR * Integrity check failed"<<endl;
			cout<<"Left child parent mismatch"<<endl;
			exit(1);
			}
			_check_integrity(r);
			x = _left(x);
			}
			}


			public:

			typedef Tree_iterator<value_type> iterator;
			typedef Tree_const_iterator<value_type> const_iterator;

			typedef std::reverse_iterator<iterator> reverse_iterator;
			typedef std::reverse_iterator<const_iterator> const_reverse_iterator;

			typedef Tree_leaf_iterator<value_type> leaf_iterator;
			typedef Tree_leaf_const_iterator<value_type> const_leaf_iterator;



			// allocation/deallocation
			Tree() {}

			Tree(const cost_functor& cost, const allocator_type& a = allocator_type())
			: impl_(cost, _Node_allocator(a)) { }

			Tree(const Tree& x) : impl_(x.impl_.cost_, x._get_Node_allocator())
			{
			if (x._root() != 0)
			{
			_root() = _copy(x._begin(), _end());
			_leftmost() = Tree_node_base::_minimum(_root());
			_rightmost() = Tree_node_base::_maximum(_root());
			impl_.node_count_ = x.impl_.node_count_;
			}
			}

			//! construct a tree from link type, defined by the user in any way
			Tree(link_type x) {

			// integrity check
			_check_integrity(x);
			if (x) _insert(0,x);
			}


			~Tree() noexcept
			{ _erase(_begin()); }

			const cost_functor& cost() const
			{ return impl_.cost_; }

			void clear() noexcept
			{
			_erase(_begin());
			_leftmost() = _end();
			_root() = 0;
			_rightmost() = _end();
			// assert count == 0, _erase should take care of decrementing the node count
			assert(impl_.node_count_ == 0);
			// impl_.node_count_ = 0;
			}

			iterator begin()
			{ return iterator(static_cast<link_type>(this->impl_.header_.left_)); }

			const_iterator begin() const
			{ return const_iterator(static_cast<const_link_type>(this->impl_.header_.left_)); }

			iterator end()
			{ return iterator(static_cast<link_type>(&this->impl_.header_)); }

			const_iterator end() const
			{ return const_iterator(static_cast<const_link_type>(&this->impl_.header_)); }


			reverse_iterator rbegin()
			{ return reverse_iterator(end()); }

			const_reverse_iterator rbegin() const
			{ return const_reverse_iterator(end()); }

			reverse_iterator rend()
			{ return reverse_iterator(begin()); }

			const_reverse_iterator rend() const
			{ return const_reverse_iterator(begin()); }


			leaf_iterator leaves_begin()
			{ return leaf_iterator(static_cast<link_type>(impl_.header_.left_)); }

			const_leaf_iterator leaves_begin() const
			{ return const_leaf_iterator(static_cast<const_link_type>(impl_.header_.left_)); }

			leaf_iterator leaves_end()
			{ return leaf_iterator(static_cast<link_type>(&impl_.header_)); }

			const_leaf_iterator leaves_end() const
			{ return const_leaf_iterator(static_cast<const_link_type>(&impl_.header_)); }

			iterator root()
			{ return iterator(static_cast<link_type>(impl_.header_.parent_)); }

			const_iterator root() const
			{ return const_iterator(static_cast<const_link_type>(impl_.header_.parent_)); }

			bool empty() const
			{ return impl_.node_count_ == 0; }

			size_type size() const
			{ return impl_.node_count_; }

			size_type max_size() const
			{ return _get_Node_allocator().max_size(); }


			std::tuple<size_t, size_t, size_t> height() const {

			size_t count = 0;
			size_t min = std::numeric_limits<size_t>::max();
			size_t max = 0;
			size_t h = 0;
			for (const_leaf_iterator it = leaves_begin(); it != leaves_end(); ++it) {
			const_base_ptr x = it.node_;
			size_t h1 = 0;
			while (x->parent_ != _end()) {
			++h1;
			x = x->parent_;
			}
			min = std::min(min, h1);
			max = std::max(max, h1);
			++count;
			h += h1;
			}

			return std::make_tuple(h/count, min, max);
			}


			private:

			void _add_level(std::queue<link_type>& q, base_ptr x, size_t h, size_t level) {

			if (h < level) {
			if (x->left_ != 0) _add_level(q, x->left_, h+1, level);
			if (x->right_ != 0) _add_level(q, x->right_, h+1, level);
			}
			else if (h == level)
			q.push(static_cast<link_type>(x));
			else
			return;
			}



			iterator _insert(base_ptr l, base_ptr r, bool count = true) {

			#ifdef DEBUG_TREE
			cout<<"inside inserting two for iterators"<<endl;
			cout<<"left pointer "<<l<<endl;
			if (l)
			cout<<"left "<<_value(l)<<endl;
			cout<<"right pointer "<<r<<endl;
			cout<<"right "<<_value(r)<<endl;
			#endif

			// set parent to header to count nodes
			link_type p = _end();
			r->parent_ = p;

			// get left most
			base_ptr lt = r;
			while (lt->left_ != 0)
			lt = lt->left_;

			#ifdef DEBUG_TREE
			cout<<"left most -> "<<_value(lt)<<endl;
			#endif

			// count nodes
			if (count)
			for (iterator it(static_cast<link_type>(lt)); it != end(); ++it)
			++impl_.node_count_;

			// get right most
			base_ptr rt = r;
			while (rt->right_ != 0)
			rt = rt->right_;

			// if l is not null, create a new node
			link_type lr = static_cast<link_type>(r);
			if (l) {

			auto vp = impl_.cost_(_value(l),_value(r));
			lr = _create_node(vp);
			++impl_.node_count_;

			p = _parent(l);
			lr->parent_ = p;
			lr->right_ = r;
			lr->left_ = l;

			r->parent_ = lr;
			l->parent_ = lr;

			} else
			_leftmost() = lt;

			// fix header
			if (lr->parent_ == _end()) {

			_root() = lr;
			_rightmost() = rt;

			} else {
			// fix parent pointers
			if (p->right_ == l)
			p->right_ = lr;
			else if (p->left_ == l)
			p->left_ = lr;
			}

			// recursively fix parents
			auto fix = _value(lr);
			while (lr->parent_ != _end()) {
			p = _parent(lr);
			fix = impl_.cost_(_value(p), fix);
			p->value_ = fix;
			lr = p;
			}

			return iterator(static_cast<link_type>(r));
			}

			bool _is_leaf(const_base_ptr x) const
			{ assert(x != nullptr); return (_right(x) == nullptr) && (_left(x) == nullptr); }



			public:


			struct _Queue_compare {

			// pointer - ancestor expansion - best cost tuples
			typedef std::tuple<link_type, Real, Real> tuple_type;

			bool operator()(const tuple_type& t1, const tuple_type& t2) const
			{ return std::get<1>(t1) > std::get<1>(t2); }
			};

			//! Adapted from Omohundro:89 routine best_sib
			template <typename Arg>
			iterator best_sibling(Arg&& v) {

			#ifdef DEBUG_TREE
			cout<<"--------------------"<<endl;
			cout<<"inserting "<<v<<endl;
			#endif

			typedef typename _Queue_compare::tuple_type tuple_type;
			typedef typename std::priority_queue<tuple_type, std::vector<tuple_type>, _Queue_compare> queue_type;

			// set result to root
			link_type result = _begin();

			// check for empty tree
			if (!result)
			return iterator(nullptr);

			// check for leaf node
			else if (_is_leaf(result))
			return iterator(result);

			// compute combined cost
			Real bcost = impl_.cost_(impl_.cost_(v, _value(result)));

			#ifdef DEBUG_TREE
			cout<<"root -> "<<_value(result)<<endl;
			cout<<"bcost -> "<<bcost<<endl;
			#endif

			// start the queue
			queue_type q;
			decltype(impl_.cost_(v)) e, c;

			q.push(std::make_tuple(result, 0., bcost));

			// while queue is not empty
			while (!q.empty()) {

			// best candidate
			auto t = q.top();
			auto nd = std::get<0>(t);
			auto aexp = std::get<1>(t);
			auto ndvol = std::get<2>(t);
			// remove top most element
			q.pop();

			#ifdef DEBUG_TREE
			cout<<" inside while"<<endl;
			cout<<" top queue item "<<_value(nd)<<endl;
			#endif

			// no way to get better than bnd
			if (aexp >= bcost) {
			#ifdef DEBUG_TREE
			cout<<"*** BREAKING"<<endl;
			#endif
			break;
			}

			// new ancestor expansion
			e = aexp + ndvol - impl_.cost_(_value(nd));

			#ifdef DEBUG_TREE
			cout<<" a exp "<<e<<endl;
			cout<<" bcost "<<bcost<<endl;
			cout<<" cost x "<<impl_.cost_(_value(nd))<<endl;
			cout<<" e -> "<<e<<endl;
			#endif

			// do left node
			c = impl_.cost_(impl_.cost_(v, _value(nd->left_)));
			#ifdef DEBUG_TREE
			cout<<" c left -> "<<c<<endl;
			#endif
			if (c+e <= bcost) {
			bcost = c+e;
			result = _left(nd);
			#ifdef DEBUG_TREE
			cout<<" c+e < bcost!!! result set to left "<<_value(result)<<endl;
			#endif
			}
			if(!_is_leaf(nd->left_))
			q.push(std::make_tuple(_left(nd), e, c));

			// do right node
			c = impl_.cost_(impl_.cost_(v, _value(nd->right_)));
			#ifdef DEBUG_TREE
			cout<<" c right -> "<<c<<endl;
			#endif
			if (c+e <= bcost) {
			bcost = c+e;
			result = _right(nd);
			#ifdef DEBUG_TREE
			cout<<" c+e < bcost!!! result set to right "<<_value(result)<<endl;
			#endif
			}
			if(!_is_leaf(nd->right_))
			q.push(std::make_tuple(_right(nd), e, c));

			} // while queue is not empty

			#ifdef DEBUG_TREE
			cout<<"best sibling -> "<<_value(result)<<endl;
			#endif
			return iterator(result);
			}

			//! Adapted from Omohundro:89 routine cheap_best_sib
			template <typename Arg>
			iterator cheap_best_sibling(Arg&& v) {

			#ifdef DEBUG_TREE
			cout<<"--------------------"<<endl;
			cout<<"inserting "<<v<<endl;
			#endif

			link_type x = _begin();

			if (!x)
			return iterator(nullptr);

			#ifdef DEBUG_TREE
			cout<<"root -> "<<_value(x)<<endl;
			#endif

			link_type result = x;

			// compute combined cost
			Real wv = impl_.cost_(v, _value(x));
			// node vol + ancestor expansion
			Real bcost = wv;
			// ancestor expansion starts at zero
			Real ae = 0;

			#ifdef DEBUG_TREE
			cout<<"bcost -> "<<bcost<<endl;
			#endif

			// while not a leaf node
			while (x->left_ && x->right_) {

			#ifdef DEBUG_TREE
			cout<<" inside while"<<endl;
			#endif
			auto nd = impl_.cost_(_value(x));

			// correct for both children
			ae += wv - nd;

			#ifdef DEBUG_TREE
			cout<<" top queue item "<<_value(x)<<endl;
			cout<<" a exp "<<ae<<endl;
			cout<<" bcost "<<bcost<<endl;
			cout<<" cost x "<<nd<<endl;
			cout<<" e -> "<<ae<<endl;
			#endif

			// cannot do any better, insert the volumes in the tree
			if (ae >= bcost)
			break;
			else {

			auto lv = impl_.cost_(v, _value(x->left_));
			auto rv = impl_.cost_(v, _value(x->right_));

			#ifdef DEBUG_TREE
			cout<<" c left -> "<<lv<<endl;
			#endif
			if (ae + lv <= bcost) {
			bcost = ae + lv;
			result = _left(x);
			#ifdef DEBUG_TREE
			cout<<" c+e < bcost!!! result set to left "<<_value(_left(x))<<endl;
			#endif
			}

			#ifdef DEBUG_TREE
			cout<<" c right -> "<<rv<<endl;
			#endif

			if (ae + rv <= bcost) {
			bcost = ae + rv;
			result = _right(x);
			#ifdef DEBUG_TREE
			cout<<" c+e < bcost!!! result set to right "<<_value(_right(x))<<endl;
			#endif
			}

			// if left branch expands less
			if (lv - impl_.cost_(_value(x->left_)) <=
			rv - impl_.cost_(_value(x->right_))) {
			wv = lv;
			x = _left(x);
			} else {
			wv = rv;
			x = _right(x);
			}
			}
			} // while loop

			#ifdef DEBUG_TREE
			cout<<"best sibling -> "<<_value(result)<<endl;
			#endif
			return iterator(result);
			}

			private:

			template <class functor_type>
			void _execute_at_leaves(iterator x, std::set<base_ptr>& visited, functor_type& fn) {

			auto it = visited.find(x.node_);
			if (it != visited.end())
			return;

			visited.insert(x.node_);

			if (x.is_leaf())
			fn(x);
			else {
			if (x.node_->left_ != 0) _execute_at_leaves(iterator(static_cast<link_type>(x.node_->left_)), visited, fn);
			if (x.node_->right_ != 0) _execute_at_leaves(iterator(static_cast<link_type>(x.node_->right_)), visited, fn);
			}
			}

			public:

			//! Collect neighbors
			/*! This function is used to collect neighbors from a tree given a functor.
			* The functor should implement a parameterless and a one-parameter operator() overloads.
			* The first load is used as the predicate to finish the search of neighbors, whereas
			* the second one is executed at leaves.
			*/
			template <class functor_type>
			void collect_neighbors(iterator it, functor_type& fn) {

			// check for null ptr
			if (it.node_ == nullptr)
			return;

			std::set<base_ptr> visited;

			base_ptr p = it.node_;

			// execute while predicate is not true
			while (!fn()) {

			p = p->parent_;

			// root node
			if (p == _root()->parent_)
			break;

			_execute_at_leaves(iterator(static_cast<link_type>(p)), visited, fn);
			}
			}

			template <typename Arg>
			std::pair<iterator,bool> cheap_insert(Arg&& v) {

			iterator it = cheap_best_sibling(v);
			link_type r = _create_node(v);

			return std::pair<iterator, bool>(_insert(it.node_, r), true);
			}


			template <typename Arg>
			std::pair<iterator,bool> insert(Arg&& v) {

			iterator it = best_sibling(v);
			link_type r = _create_node(v);

			return std::pair<iterator, bool>(_insert(it.node_, r), true);
			}


			std::pair<iterator,bool> insert(link_type x)
			{ return insert(iterator(x)); }

			std::pair<iterator,bool> insert(iterator vit) {

			iterator it = best_sibling(_value(vit.node_));
			return std::pair<iterator, bool>(_insert(it.node_, vit.node_), true);
			}


			iterator sibling(iterator it) {

			base_ptr x = it.node_;
			if (!x)
			return iterator(nullptr);
			if (x == _root())
			return end();

			base_ptr p = x->parent_;
			assert (p != 0);
			if (p->left_ == x)
			return iterator(_right(p));
			return iterator(_left(p));
			}

			public:

			iterator relocate_parent(iterator it) {

			#if DEBUG_TREE
			if (it.node_)
			cout<<"relocating node "<<_value(it.node_)<<endl;
			#endif

			link_type x = static_cast<link_type>(it.node_);


			// null pointer
			if (!x)
			return end();

			// get parent
			link_type p = _parent(x);

			// if parent is header return root
			if (p == _end())
			return iterator(_begin());

			// get grand-parent
			link_type gp = _parent(p);

			// no need to recompute volume as the parent has
			// the enclosing volume of its children

			// sibling
			link_type s = x == p->left_ ? _right(p) : _left(p);

			#ifdef DEBUG_TREE
			cout<<"relocating x -> "<<_value(x)<<endl;
			cout<<"parent of node -> "<<_value(p)<<endl;
			cout<<"sibling -> "<<_value(s)<<endl;
			#endif

			if (gp == _end()) {
			#ifdef DEBUG_TREE
			cout<<"* WARNING * sibling new parent is the header!"<<endl;
			#endif
			_root() = s;
			} else if (gp->left_ == p) {
			gp->left_ = s;
			#ifdef DEBUG_TREE
			cout<<"setting parent->parent->left to sibling"<<endl;
			#endif
			}
			else if (gp->right_ == p) {
			gp->right_ = s;
			#ifdef DEBUG_TREE
			cout<<"setting parent->parent->right to sibling"<<endl;
			#endif
			} else
			assert(false);

			// set parent of sibling
			s->parent_ = p->parent_;

			// recursively fix parents
			link_type r = _parent(s);
			while (r != _end()) {
			r->value_ = impl_.cost_(_value(r->right_), _value(r->left_));
			r = _parent(r);
			}

			// destroy parent node p and reinsert x into the tree
			x->parent_ = 0;
			p->parent_ = 0;
			p->left_ = 0;
			p->right_ = 0;
			_erase(p);

			// get the best sibling
			iterator bit = best_sibling(_value(x));

			#ifdef DEBUG_TREE
			cout<<"best sibling for relocation -> "<<bit.node_<<endl;
			if (it.node_)
			cout<<" best sibling value -> "<<_value(bit.node_)<<endl;
			#endif

			iterator result = _insert(bit.node_, x, false);

			// set leftmost and rightmost
			base_ptr lt = _root();
			while (lt->left_ != 0)
			lt = lt->left_;
			_leftmost() = lt;

			base_ptr rt = _root();
			while (rt->right_ != 0)
			rt = rt->right_;
			_rightmost() = rt;

			return result;
			}


			iterator erase(iterator it) {

			link_type x = static_cast<link_type>(it.node_);

			// null pointer
			if (!x)
			return end();

			// root node
			else if (x == _root()) {
			clear();
			return end();
			}

			link_type p = _parent(x);

			link_type s; // sibling
			if (x == p->left_) {
			s = _right(p);
			p->right_ = 0; // set to zero to use erase
			} else {
			s = _left(p);
			p->left_ = 0;
			}

			#ifdef DEBUG_TREE
			cout<<"erasing x -> "<<_value(x)<<endl;
			cout<<"parent of node -> "<<_value(p)<<endl;
			cout<<"sibling -> "<<_value(s)<<endl;
			#endif

			if (_leftmost() == x) {
			base_ptr l = s;
			while (l->left_ != 0)
			l = l->left_;
			_leftmost() = l;
			#ifdef DEBUG_TREE
			cout<<"x is leftmost, so setting leftmost to "<<_value(l)<<endl;
			#endif
			} if (_rightmost() == x) {
			base_ptr r = s;
			while (r->right_ != 0)
			r = r->right_;
			_rightmost() = r;
			#ifdef DEBUG_TREE
			cout<<"x is rightmost, so setting rightmost to "<<_value(r)<<endl;
			#endif
			}

			s->parent_ = p->parent_;
			if (p->parent_ == _end()) {
			#ifdef DEBUG_TREE
			cout<<"* WARNING * sibling new parent is the header!"<<endl;
			#endif
			_root() = s;
			} else if (p->parent_->left_ == p) {
			p->parent_->left_ = s;
			#ifdef DEBUG_TREE
			cout<<"setting parent->parent->left to sibling"<<endl;
			#endif
			}
			else {
			p->parent_->right_ = s;
			#ifdef DEBUG_TREE
			cout<<"setting parent->parent->right to sibling"<<endl;
			#endif
			}


			#ifdef DEBUG_TREE
			cout<<"p right -> "<<p->right_<<endl;
			cout<<"p left -> "<<p->left_<<endl;
			if (p->right_)
			cout<<"p right value -> "<<_value(p->right_)<<endl;
			if (p->left_)
			cout<<"p left value -> "<<_value(p->left_)<<endl;
			#endif

			// recursively fix parents
			link_type r = _parent(s);
			while (r != _end()) {
			r->value_ = _value(r->right_) + _value(r->left_);
			r = _parent(r);
			}

			// erase subtree rooted at p
			_erase(p);

			return iterator(s);
			}



			Real volume() const {
			Real v = 0;
			for (const_iterator it = begin(); it != end(); ++it)
			v += it->measure();
			return v;
			}


			friend std::ostream& operator<<(std::ostream& os, const Tree& t) {

			os<<"Tree: count "<<t.impl_.node_count_<<endl;
			os<<" header: "<<&t.impl_.header_<<endl;
			if (t.impl_.header_.parent_)
			os<<" ROOT: "<<t.impl_.header_.parent_<<", value "<<_value(t.impl_.header_.parent_)<<endl;
			os<<" left: "<<t.impl_.header_.left_<<endl;
			os<<" right: "<<t.impl_.header_.right_<<endl;
			size_t c = 0;
			Real v = 0;
			os<<endl;
			for (Tree::const_iterator it = t.begin(); it != t.end(); ++it, ++c) {

			Real cost = t.impl_.cost_(*it);

			if (it.node_ == t.impl_.header_.parent_) {
			os<<" ROOT "<<it.node_;
			if (it.node_ == t._leftmost())
			os<<" [LEFTMOST]";
			if (it.node_ == t._rightmost())
			os<<" [RIGHTMOST]";
			os<<"\n value: "<<*it<<endl;
			os<<" parent: "<<it.node_->parent_<<endl;
			} else {
			os<<" pointer "<<it.node_;
			if (it.node_ == t._leftmost())
			os<<" [LEFTMOST]";
			if (it.node_ == t._rightmost())
			os<<" [RIGHTMOST]";
			os<<"\n value: "<<*it<<endl;
			os<<" parent: "<<it.node_->parent_<<", value "<<_value(it.node_->parent_)<<endl;
			}

			if (it.node_->left_)
			os<<" left: "<<it.node_->left_<<", value "<<_value(it.node_->left_)<<endl;
			if (it.node_->right_)
			os<<" right: "<<it.node_->right_<<", value "<<_value(it.node_->right_)<<endl;
			if (!it.node_->left_ && !it.node_->right_)
			os<<" leaf"<<endl;
			os<<" cost: "<<cost<<endl;
			os<<endl;

			v += cost;
			}
			cout<<"total cost -> "<<v<<endl;

			if (c != t.impl_.node_count_)
			cout<<"* WARNING * Tree node count mismatch "<<t.impl_.node_count_<<" != "<<c<<endl;
			return os;
			}

			template <class functor_type>
			functor_type execute_at_level(size_t level, functor_type fn) {

			link_type x = _begin();
			if (x == nullptr)
			return fn;

			std::queue<link_type> q;
			_add_level(q, x, 0, level);

			while (!q.empty()) {
			link_type l = q.front();
			assert(l != 0);
			fn(iterator(l));
			q.pop();
			}
			return fn;
			}
			};





			template <class object_type>
			struct Cost_functor {

			auto operator()(const object_type& o) const -> decltype(o.measure())
			{ return o.measure(); }

			auto operator()(const object_type& o1, const object_type& o2) const -> decltype(o1+o2)
			{ return (o1+o2); }

			};


			template <class link_type>
			struct Tuple_compare {

			typedef std::tuple<link_type, link_type, Real> tuple_type;

			bool operator()(const tuple_type& t1, const tuple_type& t2) const
			{ return std::get<2>(t1) > std::get<2>(t2); }
			};

			// bottom up tree construction
			template <class tree_type, class volume_container>
			tree_type* construct_tree_bottom_up(const volume_container& volumes)
			{
			// online tree type definitions
			typedef typename tree_type::value_type volume_type;
			typedef typename tree_type::cost_functor cost_functor;
			typedef typename tree_type::iterator iterator_type;

			// queue type definitions
			typedef std::tuple<iterator_type, iterator_type, Real> tuple_type;
			typedef typename std::priority_queue<tuple_type, std::vector<tuple_type>, Tuple_compare<iterator_type> > queue_type;

			// already paired container type definitions
			typedef std::set<iterator_type> paired_set;
			typedef typename paired_set::iterator paired_iterator;

			size_t numVolumes = volumes.size();
			assert(numVolumes != 0);

			tree_type* tp = new tree_type();
			const cost_functor& cost = tp->cost();
			tree_type& t = *tp;
			queue_type q;
			paired_set s;

			std::vector<iterator_type> links(numVolumes);

			// insert volumes into online tree
			for (size_t i=0; i<numVolumes; ++i) {
			std::pair<iterator_type, bool> p = t.insert(volumes[i]);
			assert(p.second);
			links[i] = p.first;
			}

			iterator_type ii,jj;
			for (size_t i=0; i<numVolumes-1; ++i) {

			Real merged = std::numeric_limits<Real>::infinity();

			for (size_t j=i+1; j<numVolumes; ++j) {

			volume_type v = cost(links[i], links[j]);
			Real measure = cost(v);
			if (measure < merged) {
			ii = links[i];
			jj = links[j];
			merged = measure;
			}
			}
			q.push(std::make_tuple(ii,jj,merged));
			}

			while (!q.empty()) {

			// best candidate
			auto tuple = q.top();
			iterator_type node = std::get<0>(tuple);
			iterator_type pair = std::get<1>(tuple);

			q.pop();

			paired_iterator it = s.find(node);
			if (it != s.end())
			continue;

			it = s.find(pair);
			if (it != s.end())
			continue;

			#if DEBUG_TREE
			cout<<t<<endl;
			cout<<"node -> "<<node.node_<<endl;
			cout<<"node value -> "<<*node<<endl;
			cout<<"pair -> "<<pair.node_<<endl;
			cout<<"pair value -> "<<*pair<<endl;
			cout<<"volume -> "<<std::get<2>(tuple)<<endl;

			#endif

			// obtain best pair from online tree
			iterator_type best = t.sibling(node);
			#if DEBUG_TREE
			cout<<"best sibling -> "<<*best<<endl;
			#endif
			if (pair == best) {

			iterator_type parent = t.relocate_parent(node);

			assert(parent != t.end());

			#if DEBUG_TREE
			cout<<"BEST SIBLING MATCHES! relocating..."<<endl;
			cout<<" parent -> "<<*parent<<endl;
			#endif

			// add paired nodes to container
			s.insert(node);
			s.insert(pair);

			// compute best pair for the parent
			best = t.sibling(parent);

			if (best != t.end()) {

			tuple_type tuple = std::make_tuple(parent, best, cost(cost(parent, best)));
			q.push(tuple);

			iterator_type node = std::get<0>(tuple);
			iterator_type pair = std::get<1>(tuple);

			#if DEBUG_TREE
			cout<<" pushing new tuple "<<endl;
			cout<<" node -> "<<*node<<endl;
			cout<<" pair -> "<<*pair<<endl;
			cout<<" volume -> "<<std::get<2>(tuple)<<endl;
			#endif
			}
			}
			// else enqueue new pair
			else {

			if (best != t.end()) {

			tuple_type tuple = std::make_tuple(node,best, cost(cost(node, best)));
			q.push(tuple);

			iterator_type node = std::get<0>(tuple);
			iterator_type pair = std::get<1>(tuple);

			#if DEBUG_TREE
			cout<<"BEST SIBLING DOES NOT MATCH!"<<endl;
			cout<<" pushing new tuple "<<endl;
			cout<<" node -> "<<*node<<endl;
			cout<<" pair -> "<<*pair<<endl;
			cout<<" volume -> "<<std::get<2>(tuple)<<endl;
			#endif
			}

			}
			}

			return tp;
			}



			template <class volume_type>
			class Volume_creator;

			template<int d>
			struct Volume_creator<Ball<d> > {

			typedef typename Ball<d>::point_type point_type;

			template <class coord_array>
			static Ball<d> create(const coord_array& coord) {

			std::vector<point_type> pts;
			size_t nnodes = coord.size();
			pts.reserve(nnodes);
			for (size_t i=0; i<nnodes; ++i)
			pts.push_back(point_type(coord[i]));
			return bounding_ball<d>(pts);
			}
			};

			template <int d>
			struct Volume_creator<BoundingBox<d> > {

			typedef typename BoundingBox<d>::point_type point_type;

			template <class coord_array>
			static Sphere create(const coord_array& coord) {

			std::vector<point_type> pts;
			size_t nnodes = coord.size();
			pts.reserve(nnodes);
			for (size_t i=0; i<nnodes; ++i)
			pts.push_back(point_type(coord[i]));
			return BoundingBox<d>(pts.begin(), pts.end());
			}
			};



			// bottom up tree construction
			template <class tree_type, class model_type, class element_type>
			tree_type* construct_tree_bottom_up(model_type& model) {

			// online tree type definitions
			typedef typename tree_type::value_type volume_type;
			typedef typename tree_type::cost_functor cost_functor;
			typedef typename tree_type::iterator iterator_type;

			// queue type definitions
			typedef std::tuple<iterator_type, iterator_type, Real> tuple_type;
			typedef typename std::priority_queue<tuple_type, std::vector<tuple_type>, Tuple_compare<iterator_type> > queue_type;

			// already paired container type definitions
			typedef std::set<iterator_type> paired_set;
			typedef typename paired_set::iterator paired_iterator;

			tree_type* tp = new tree_type();
			tree_type& t = *tp;
			const cost_functor& cost = t.cost();
			std::vector<iterator_type> links;

			typedef typename model_type::mesh_type mesh_type;

			mesh_type& mesh = model.getMesh();
			int dim = mesh.getSpatialDimension();

			// iterate over elements of lower dimension
			typename mesh_type::type_iterator it = mesh.firstType(dim-1);
			typename mesh_type::type_iterator end = mesh.lastType(dim-1);

			size_t numVolumes = 0;
			for(; it != end; ++it) {

			// add elements to corresponding surface
			for(UInt e = 0; e < mesh.getNbElement(*it); ++e) {

			// create solid mechanics element
			element_type el(model, *it, e);

			// vector of coordinates for update
			std::vector<const Real*> coord = el.coordinates();

			// create leaf volume
			volume_type v = Volume_creator<volume_type>::create(coord);

			// do not add zero measure volumes
			if (v.measure() == 0 && dim != 1)
			continue;

			std::pair<iterator_type, bool> p = t.insert(v);
			assert(p.second);
			links.push_back(p.first);
			++numVolumes;

			// save leaf data
			if (!t.add_data(p.first,el)) {
			cout<<"* ERROR * Could not add data. Aborting..."<<endl;
			exit(1);
			}


			} // loop over elements
			} // loop over types of element

			if (numVolumes == 0) {
			cout<<"* ERROR * No volumes were created. Aborting..."<<endl;
			exit(1);
			}

			queue_type q;
			paired_set s;

			iterator_type ii,jj;
			for (size_t i=0; i<numVolumes-1; ++i) {

			Real merged = std::numeric_limits<Real>::infinity();

			for (size_t j=i+1; j<numVolumes; ++j) {

			volume_type v = cost(links[i], links[j]);
			Real measure = v.measure();
			if (measure < merged) {
			ii = links[i];
			jj = links[j];
			merged = measure;
			}
			}
			q.push(std::make_tuple(ii,jj,merged));
			}

			while (!q.empty()) {

			// best candidate
			auto tuple = q.top();
			iterator_type node = std::get<0>(tuple);
			iterator_type pair = std::get<1>(tuple);

			q.pop();

			paired_iterator it = s.find(node);
			if (it != s.end())
			continue;

			it = s.find(pair);
			if (it != s.end())
			continue;

			#if DEBUG_TREE
			cout<<t<<endl;
			cout<<"node -> "<<node.node_<<endl;
			cout<<"node value -> "<<*node<<endl;
			cout<<"pair -> "<<pair.node_<<endl;
			cout<<"pair value -> "<<*pair<<endl;
			cout<<"volume -> "<<std::get<2>(tuple)<<endl;

			#endif

			// obtain best pair from online tree
			iterator_type best = t.sibling(node);
			#if DEBUG_TREE
			cout<<"best sibling -> "<<*best<<endl;
			#endif
			if (pair == best) {

			iterator_type parent = t.relocate_parent(node);

			assert(parent != t.end());

			#if DEBUG_TREE
			cout<<"BEST SIBLING MATCHES! relocating..."<<endl;
			cout<<" parent -> "<<*parent<<endl;
			#endif

			// add paired nodes to container
			s.insert(node);
			s.insert(pair);

			// compute best pair for the parent
			best = t.sibling(parent);

			if (best != t.end()) {

			tuple_type tuple = std::make_tuple(parent, best, cost(cost(parent,best)));
			q.push(tuple);

			iterator_type node = std::get<0>(tuple);
			iterator_type pair = std::get<1>(tuple);

			#if DEBUG_TREE
			cout<<" pushing new tuple "<<endl;
			cout<<" node -> "<<*node<<endl;
			cout<<" pair -> "<<*pair<<endl;
			cout<<" volume -> "<<std::get<2>(tuple)<<endl;
			#endif
			}
			}
			// else enqueue new pair
			else {

			if (best != t.end()) {

			tuple_type tuple = std::make_tuple(node,best, cost(cost(node, best)));
			q.push(tuple);

			iterator_type node = std::get<0>(tuple);
			iterator_type pair = std::get<1>(tuple);

			#if DEBUG_TREE
			cout<<"BEST SIBLING DOES NOT MATCH!"<<endl;
			cout<<" pushing new tuple "<<endl;
			cout<<" node -> "<<*node<<endl;
			cout<<" pair -> "<<*pair<<endl;
			cout<<" volume -> "<<std::get<2>(tuple)<<endl;
			#endif
			}

			}
			}

			return tp;
			}


			// bottom up tree construction
			template <class tree_type, class element_container>
			tree_type* construct_tree_bottom_up(element_container& elems) {

			// element type
			typedef typename element_container::value_type element_type;

			// online tree type definitions
			typedef typename tree_type::value_type volume_type;
			typedef typename tree_type::cost_functor cost_functor;
			typedef typename tree_type::iterator iterator_type;

			// queue type definitions
			typedef std::tuple<iterator_type, iterator_type, Real> tuple_type;
			typedef typename std::priority_queue<tuple_type, std::vector<tuple_type>, Tuple_compare<iterator_type> > queue_type;

			// already paired container type definitions
			typedef std::set<iterator_type> paired_set;
			typedef typename paired_set::iterator paired_iterator;

			tree_type* tp = new tree_type();
			tree_type& t = *tp;
			const cost_functor& cost = t.cost();
			std::vector<iterator_type> links;

			size_t numVolumes = 0;

			for (typename element_container::iterator it = elems.begin();
			it != elems.end(); ++it) {

			element_type &el = *it;

			// vector of coordinates for update
			std::vector<const Real*> coord = el.coordinates();

			// create leaf volume
			volume_type v = Volume_creator<volume_type>::create(coord);

			// do not add zero measure volumes
			if (v.measure() == 0)
			continue;

			std::pair<iterator_type, bool> p = t.insert(v);
			assert(p.second);
			links.push_back(p.first);
			++numVolumes;

			// save leaf data
			if (!t.add_data(p.first,el)) {
			cout<<"* ERROR * Could not add data. Aborting..."<<endl;
			exit(1);
			}

			} // loop over elements in container

			if (numVolumes == 0) {
			cout<<"* ERROR * No volumes were created. Aborting..."<<endl;
			exit(1);
			}

			queue_type q;
			paired_set s;

			iterator_type ii,jj;
			for (size_t i=0; i<numVolumes-1; ++i) {

			Real merged = std::numeric_limits<Real>::infinity();

			for (size_t j=i+1; j<numVolumes; ++j) {

			volume_type v = cost(links[i], links[j]);
			Real measure = v.measure();
			if (measure < merged) {
			ii = links[i];
			jj = links[j];
			merged = measure;
			}
			}
			q.push(std::make_tuple(ii,jj,merged));
			}

			while (!q.empty()) {

			// best candidate
			auto tuple = q.top();
			iterator_type node = std::get<0>(tuple);
			iterator_type pair = std::get<1>(tuple);

			q.pop();

			paired_iterator it = s.find(node);
			if (it != s.end())
			continue;

			it = s.find(pair);
			if (it != s.end())
			continue;

			#if DEBUG_TREE
			cout<<t<<endl;
			cout<<"node -> "<<node.node_<<endl;
			cout<<"node value -> "<<*node<<endl;
			cout<<"pair -> "<<pair.node_<<endl;
			cout<<"pair value -> "<<*pair<<endl;
			cout<<"volume -> "<<std::get<2>(tuple)<<endl;

			#endif

			// obtain best pair from online tree
			iterator_type best = t.sibling(node);
			#if DEBUG_TREE
			cout<<"best sibling -> "<<*best<<endl;
			#endif
			if (pair == best) {

			iterator_type parent = t.relocate_parent(node);

			assert(parent != t.end());

			#if DEBUG_TREE
			cout<<"BEST SIBLING MATCHES! relocating..."<<endl;
			cout<<" parent -> "<<*parent<<endl;
			#endif

			// add paired nodes to container
			s.insert(node);
			s.insert(pair);

			// compute best pair for the parent
			best = t.sibling(parent);

			if (best != t.end()) {

			tuple_type tuple = std::make_tuple(parent, best, cost(cost(parent,best)));
			q.push(tuple);

			iterator_type node = std::get<0>(tuple);
			iterator_type pair = std::get<1>(tuple);

			#if DEBUG_TREE
			cout<<" pushing new tuple "<<endl;
			cout<<" node -> "<<*node<<endl;
			cout<<" pair -> "<<*pair<<endl;
			cout<<" volume -> "<<std::get<2>(tuple)<<endl;
			#endif
			}
			}
			// else enqueue new pair
			else {

			if (best != t.end()) {

			tuple_type tuple = std::make_tuple(node,best, cost(cost(node, best)));
			q.push(tuple);

			iterator_type node = std::get<0>(tuple);
			iterator_type pair = std::get<1>(tuple);

			#if DEBUG_TREE
			cout<<"BEST SIBLING DOES NOT MATCH!"<<endl;
			cout<<" pushing new tuple "<<endl;
			cout<<" node -> "<<*node<<endl;
			cout<<" pair -> "<<*pair<<endl;
			cout<<" volume -> "<<std::get<2>(tuple)<<endl;
			#endif
			}

			}
			}

			return tp;
			}




			template <class tree_type>
			void print_mathematica(tree_type& t) {

			typedef typename tree_type::value_type volume_type;

			int d = volume_type::dim();

			cout<<std::fixed;
			cout.precision(2);

			if (d == 2)
			cout<<"Graphics[{";
			else
			cout<<"Graphics3D[{";

			for (typename tree_type::const_leaf_iterator it = t.leaves_begin(); it != t.leaves_end(); ++it)
			cout<<*it<<", ";

			cout<<"}];"<<endl;
			}


			template <typename T>
			std::string stringify(const T& x) {
			std::stringstream o;
			if (!(o << x)) {
			cout<<"* ERROR * Bad argument to stringity function"<<endl;
			exit(1);
			}
			return o.str();
			}

			template <class tree>
			void export_video(tree& t, Real duration, Real min_opacity, std::string pre = "", std::string post = "") {

			typedef typename tree::value_type volume_type;
			typedef typename volume_type::aabb_type aabb_type;
			typedef typename aabb_type::point_type point_type;
			typedef typename point_type::value_type value_type;
			typedef typename tree::iterator iterator;

			int d = volume_type::dim();
			cout<<std::fixed;
			cout.precision(2);

			std::tuple<size_t, size_t, size_t> h = t.height();

			typename tree::iterator it = t.begin();

			aabb_type bb = it->bounding_box();

			// get bounding box
			for (; it != t.end(); ++it)
			bb += it->bounding_box();

			std::string figs;
			int H = std::get<2>(h);
			int k = 0;
			for (int l = H; l>=0; --l) {

			std::string g = "g";
			g += stringify(k);
			++k;

			// compute opacity for figure
			Real opacity = (1 - min_opacity)*static_cast<double>(l)/H + min_opacity;

			if (d == 2)
			cout<<g<<" = Graphics[{Opacity["<<stringify(opacity)<<"],"<<pre;
			else
			cout<<g<<" = Graphics3D[{Opacity["<<stringify(opacity)<<"],"<<pre;

			t.execute_at_level(l, [](iterator it){ cout<<*it<<", ";});

			figs += figs.empty() ? g : (','+g);

			cout<<"}];\nShow["<<figs<<", PlotRange-> {{";
			const point_type &m = bb.min();
			const point_type &M = bb.max();
			for (int i=0; i<d; ++i) {
			value_type l = 0.05*(M[i] - m[i]);
			cout<<(m[i]-l)<<','<<(M[i]+l)<<(i < d-1 ? "},{" : "}}");
			}
			cout<<post<<"];\n";
			cout<<"Export[\""<<g<<".png\", %, ImageResolution -> 300];"<<endl;
			}

			cout<<"files = Last /@ Sort[{Characters@#, #} & /@ FileNames[\"*.png\"]]"<<endl;
			cout<<"images = Map[Import, files];"<<endl;
			cout<<"Export[\"video.mov\", images, \"FrameRate\" -> "<<stringify(static_cast<double>(k)/duration)<<"];"<<endl;
			}


			//// bottom up tree construction
			//template <class volume_container>
			//Tree<typename volume_container::value_type, Cost_functor<typename volume_container::value_type> >
			//construct_tree_bottom_up2(const volume_container& volumes)
			//{
			//
			// typedef typename volume_container::value_type volume_type;
			//
			// // online tree type definitions
			// typedef Tree<volume_type, Cost_functor<volume_type> > tree_type;
			//
			// typedef Tree_node<volume_type> node_type;
			// typedef typename tree_type::link_type link_type;
			//
			// size_t numVolumes = volumes.size();
			// assert(numVolumes != 0);
			//
			// // allocate leafs of the tree
			// link_type* leafs = new link_type[numVolumes];
			//
			// for (size_t i=0; i<numVolumes; ++i)
			// leafs[i] = new node_type(volumes[i]);
			//
			// while (numVolumes > 1) {
			//
			// // find indices of the two "nearest" nodes, based on some criterion
			// size_t ii,jj;
			// Real merged = std::numeric_limits<Real>::infinity();
			// for (size_t i=0; i<numVolumes; ++i)
			// for (size_t j=i+1; j<numVolumes; ++j) {
			//
			// volume_type v = leafs[i]->value_ + leafs[j]->value_;
			// if (v.measure() < merged) {
			// ii = i;
			// jj = j;
			// merged = v.measure();
			// }
			// }
			//
			// volume_type v = leafs[ii]->value_ + leafs[jj]->value_;
			// link_type parent = new node_type(v);
			//
			//#if DEBUG_TREE
			// cout<<"best volume found -> "<<v<<endl;
			// cout<<"measure -> "<<v.measure()<<endl;
			// cout<<"parent -> "<<parent->value_<<endl;
			// cout<<"left -> "<<leafs[ii]->value_<<endl;
			// cout<<"right -> "<<leafs[jj]->value_<<endl;
			//#endif
			//
			// parent->left_ = leafs[ii];
			// parent->right_ = leafs[jj];
			// leafs[ii]->parent_ = parent;
			// leafs[jj]->parent_ = parent;
			//
			// // remove the two nodes from the active set and add in the new node.
			// leafs[ii] = parent;
			// leafs[jj] = leafs[numVolumes - 1];
			// --numVolumes;
			// }
			//
			// link_type root = leafs[0];
			//
			// tree_type t(root);
			//
			// delete leafs;
			//
			// return t;
			//}

			__END_AKANTU__

			#endif /* __AKANTU_TREE_HH__ */

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