    // @(#)root/cont:$Name:  $:$Id: TBtree.cxx,v 1.8 2004/11/12 21:51:18 brun Exp $
    // Author: Fons Rademakers   10/10/95

    /*************************************************************************
     * Copyright (C) 1995-2000, Rene Brun and Fons Rademakers.               *
     * All rights reserved.                                                  *
     *                                                                       *
     * For the licensing terms see $ROOTSYS/LICENSE.                         *
     * For the list of contributors see $ROOTSYS/README/CREDITS.             *
     *************************************************************************/

    //////////////////////////////////////////////////////////////////////////
    //                                                                      //
    // TBtree                                                               //
    //                                                                      //
    // B-tree class. TBtree inherits from the TSeqCollection ABC.           //
    //                                                                      //
    //////////////////////////////////////////////////////////////////////////
    //

## B-tree Implementation notes

This implements B-trees with several refinements. Most of them can be found in Knuth Vol 3, but some were developed to adapt to restrictions imposed by C++. First, a restatement of Knuth's properties that a B-tree must satisfy, assuming we make the enhancement he suggests in the paragraph at the bottom of page 476. Instead of storing null pointers to non-existent nodes (which Knuth calls the leaves) we utilize the space to store keys. Therefore, what Knuth calls level (l-1) is the bottom of our tree, and we call the nodes at this level LeafNodes. Other nodes are called InnerNodes. The other enhancement we have adopted is in the paragraph at the bottom of page 477: overflow control.

The following are modifications of Knuth's properties on page 478:

1. Every InnerNode has at most Order keys, and at most Order+1 sub-trees.
2. Every LeafNode has at most 2*(Order+1) keys.
3. An InnerNode with k keys has k+1 sub-trees.
4. Every InnerNode that is not the root has at least InnerLowWaterMark keys.
5. Every LeafNode that is not the root has at least LeafLowWaterMark keys.
6. If the root is a LeafNode, it has at least one key.
7. If the root is an InnerNode, it has at least one key and two sub-trees.
8. All LeafNodes are the same distance from the root as all the other LeafNodes.
9. For InnerNode n with key n[i].key, then sub-tree n[i-1].tree contains all keys < n[i].key, and sub-tree n[i].tree contains all keys >= n[i].key.
10. Order is at least 3.

The values of InnerLowWaterMark and LeafLowWaterMark may actually be set by the user when the tree is initialized, but currently they are set automatically to:

InnerLowWaterMark = ceiling(Order/2) LeafLowWaterMark = Order - 1

If the tree is only filled, then all the nodes will be at least 2/3 full. They will almost all be exactly 2/3 full if the elements are added to the tree in order (either increasing or decreasing). [Knuth says McCreight's experiments showed almost 100% memory utilization. I don't see how that can be given the algorithms that Knuth gives. McCreight must have used a different scheme for balancing. [No, he used a different scheme for splitting: he did a two-way split instead of the three way split as we do here. Which means that McCreight does better on insertion of ordered data, but we should do better on insertion of random data.]]

It must also be noted that B-trees were designed for DISK access algorithms, not necessarily in-memory sorting, as we intend it to be used here. However, if the order is kept small (< 6?) any inefficiency is negligible for in-memory sorting. Knuth points out that balanced trees are actually preferable for memory sorting. I'm not sure that I believe this, but it's interesting. Also, deleting elements from balanced binary trees, being beyond the scope of Knuth's book (p. 465), is beyond my scope. B-trees are good enough.

A B-tree is declared to be of a certain ORDER (3 by default). This number determines the number of keys contained in any interior node of the tree. Each interior node will contain ORDER keys, and therefore ORDER+1 pointers to sub-trees. The keys are numbered and indexed 1 to ORDER while the pointers are numbered and indexed 0 to ORDER. The 0th ptr points to the sub-tree of all elements that are less than key[1]. Ptr[1] points to the sub-tree that contains all the elements greater than key[1] and less than key[2]. etc. The array of pointers and keys is allocated as ORDER+1 pairs of keys and nodes, meaning that one key field (key[0]) is not used and therefore wasted. Given that the number of interior nodes is small, that this waste allows fewer cases of special code, and that it is useful in certain of the methods, it was felt to be a worthwhile waste.

The size of the exterior nodes (leaf nodes) does not need to be related to the size of the interior nodes at all. Since leaf nodes contain only keys, they may be as large or small as we like independent of the size of the interior nodes. For no particular reason other than it seems like a good idea, we will allocate 2*(ORDER+1) keys in each leaf node, and they will be numbered and indexed from 0 to 2*ORDER+1. It does have the advantage of keeping the size of the leaf and interior arrays the same, so that if we find allocation and de-allocation of these arrays expensive, we can modify their allocation to use a garbage ring, or something.

Both of these numbers will be run-time constants associated with each tree (each tree at run-time can be of a different order). The variable "order" is the order of the tree, and the inclusive upper limit on the indices of the keys in the interior nodes. The variable "order2" is the inclusive upper limit on the indices of the leaf nodes, and is designed

(1) to keep the sizes of the two kinds of nodes the same; (2) to keep the expressions involving the arrays of keys looking somewhat the same: lower limit upper limit for inner nodes: 1 order for leaf nodes: 0 order2 Remember that index 0 of the inner nodes is special.

Currently, order2 = 2*(order+1).

Picture: (also see Knuth Vol 3 pg 478) +--+--+--+--+--+--... | | | | | | parent--->| | | | | | | | +*-+*-+*-+--+--+--... | | | +----+ | +-----+ | +-----+ | V | V +----------+ | +----------+ | | | | | this->| | | | |<--sib +----------+ | +----------+ V data

It is conceptually VERY convenient to think of the data as being the very first element of the sib node. Any primitive that tells sib to perform some action on n nodes should include this 'hidden' element. For InnerNodes, the hidden element has (physical) index 0 in the array, and in LeafNodes, the hidden element has (virtual) index -1 in the array. Therefore, there are two 'size' primitives for nodes:

Psize - the physical size: how many elements are contained in the array in the node. Vsize - the 'virtual' size; if the node is pointed to by element 0 of the parent node, then Vsize == Psize; otherwise the element in the parent item that points to this node 'belongs' to this node, and Vsize == Psize+1;

Parent nodes are always InnerNodes.

These are the primitive operations on Nodes:

Append(elt) - adds an element to the end of the array of elements in a node. It must never be called where appending the element would fill the node. Split() - divide a node in two, and create two new nodes. SplitWith(sib) - create a third node between this node and the sib node, divvying up the elements of their arrays. PushLeft(n) - move n elements into the left sibling PushRight(n) - move n elements into the right sibling BalanceWithRight() - even up the number of elements in the two nodes. BalanceWithLeft() - ditto

To allow this implementation of btrees to also be an implementation of sorted arrays/lists, the overhead is included to allow O(log n) access of elements by their rank (`give me the 5th largest element'). Therefore, each Item keeps track of the number of keys in and below it in the tree (remember, each item's tree is all keys to the RIGHT of the item's own key).

[ [ < 0 1 2 3 > 4 < 5 6 7 > 8 < 9 10 11 12 > ] 13 [ < 14 15 16 > 17 < 18 19 20 > ] ] 4 1 1 1 1 4 1 1 1 5 1 1 1 1 7 3 1 1 1 4 1 1 1

    #include <stdlib.h>
    #include "TBtree.h"

    ClassImp(TBtree)

    //______________________________________________________________________________
    TBtree::TBtree(int order)
    {
       // Create a B-tree of certain order (by default 3).

       Init(order);
    }

    //______________________________________________________________________________
    TBtree::~TBtree()
    {
       // Delete B-tree. Objects are not deleted unless the TBtree is the
       // owner (set via SetOwner()).

       if (fRoot) {
          Clear();
          SafeDelete(fRoot);
       }
    }

    //______________________________________________________________________________
    void TBtree::Add(TObject *obj)
    {
       // Add object to B-tree.

       if (IsArgNull("Add", obj)) return;
       if (!obj->IsSortable()) {
          Error("Add", "object must be sortable");
          return;
       }
       if (!fRoot) {
          fRoot = new TBtLeafNode(0, obj, this);
          Check(fRoot != 0);
          IncrNofKeys();
       } else {
          TBtNode *loc;
          Int_t idx;
          if (fRoot->Found(obj, &loc, &idx) != 0) {
             // loc and idx are set to either where the object
             // was found, or where it should go in the Btree.
             // Nothing is here now, but later we might give the user
             // the ability to declare a B-tree as `unique elements only',
             // in which case we would handle an exception here.
          }
          loc->Add(obj, idx);
       }
    }

    //______________________________________________________________________________
    TObject *TBtree::After(const TObject *) const
    {
       // Cannot use this method since B-tree decides order.

       MayNotUse("After");
       return 0;
    }

    //______________________________________________________________________________
    TObject *TBtree::Before(const TObject *) const
    {
       // May not use this method since B-tree decides order.

       MayNotUse("Before");
       return 0;
    }

    //______________________________________________________________________________
    void TBtree::Clear(Option_t *)
    {
       // Remove all objects from B-tree. Does NOT delete objects unless the TBtree
       // is the owner (set via SetOwner()).

       if (IsOwner())
          Delete();
       else {
          SafeDelete(fRoot);
          fSize = 0;
       }
    }

    //______________________________________________________________________________
    void TBtree::Delete(Option_t *)
    {
       // Remove all objects from B-tree AND delete all heap based objects.

       for (Int_t i = 0; i < fSize; i++) {
          TObject *obj = At(i);
          if (obj && obj->IsOnHeap())
             TCollection::GarbageCollect(obj);
       }
       SafeDelete(fRoot);
       fSize = 0;
    }

    //______________________________________________________________________________
    TObject *TBtree::FindObject(const char *name) const
    {
       // Find object using its name (see object's GetName()). Requires sequential
       // search of complete tree till object is found.

       return TCollection::FindObject(name);
    }

    //______________________________________________________________________________
    TObject *TBtree::FindObject(const TObject *obj) const
    {
       // Find object using the objects Compare() member function.

       if (!obj->IsSortable()) {
          Error("FindObject", "object must be sortable");
          return 0;
       }
       if (!fRoot)
          return 0;
       else {
          TBtNode *loc;
          Int_t idx;
          return fRoot->Found(obj, &loc, &idx);
       }
    }

    //______________________________________________________________________________
    Int_t TBtree::IdxAdd(const TObject &obj)
    {
       // Add object and return its index in the tree.

       Int_t r;
       if (!obj.IsSortable()) {
          Error("IdxAdd", "object must be sortable");
          return -1;
       }
       if (!fRoot) {
          fRoot = new TBtLeafNode(0, &obj, this);
          Assert(fRoot != 0);
          IncrNofKeys();
          r = 0;
       } else {
          TBtNode *loc;
          int idx;
          if (fRoot->Found(&obj, &loc, &idx) != 0) {
             // loc and idx are set to either where the object
             // was found, or where it should go in the Btree.
             // Nothing is here now, but later we might give the user
             // the ability to declare a B-tree as `unique elements only',
             // in which case we would handle an exception here.
             // cerr << "Multiple entry warning\n";
          } else {
             Check(loc->fIsLeaf);
          }
          if (loc->fIsLeaf) {
             if (loc->fParent == 0)
                r = idx;
             else
                r = idx + loc->fParent->FindRankUp(loc);
          } else {
              TBtInnerNode *iloc = (TBtInnerNode*) loc;
              r = iloc->FindRankUp(iloc->GetTree(idx));
          }
          loc->Add(&obj, idx);
       }
       Check(r == Rank(&obj) || &obj == (*this)[r]);
       return r;
    }

    //______________________________________________________________________________
    void TBtree::Init(Int_t order)
    {
       // Initialize a B-tree.

       if (order < 3) {
          Warning("Init", "order must be at least 3");
          order = 3;
       }
       fRoot   = 0;
       fOrder  = order;
       fOrder2 = 2 * (fOrder+1);
       fLeafMaxIndex  = fOrder2 - 1;     // fItem[0..fOrder2-1]
       fInnerMaxIndex = fOrder;          // fItem[1..fOrder]
       //
       // the low water marks trigger an exploration for balancing
       // or merging nodes.
       // When the size of a node falls below X, then it must be possible to
       // either balance this node with another node, or it must be possible
       // to merge this node with another node.
       // This can be guaranteed only if (this->Size() < (MaxSize()-1)/2).
       //
       //

       // == MaxSize() satisfies the above because we compare
       // lowwatermark with fLast
       fLeafLowWaterMark  = ((fLeafMaxIndex+1)-1) / 2 - 1;
       fInnerLowWaterMark = (fOrder-1) / 2;
    }

    //______________________________________________________________________________
    //void TBtree::PrintOn(ostream& out) const
    //{
    //   // Print a B-tree.
    //
    //   if (!fRoot)
    //      out << "<empty>";
    //   else
    //      fRoot->PrintOn(out);
    //}

    //______________________________________________________________________________
    TIterator *TBtree::MakeIterator(Bool_t dir) const
    {
       // Returns a B-tree iterator.

       return new TBtreeIter(this, dir);
    }

    //______________________________________________________________________________
    Int_t TBtree::Rank(const TObject *obj) const
    {
       // Returns the rank of the object in the tree.

       if (!obj->IsSortable()) {
          Error("Rank", "object must be sortable");
          return -1;
       }
       if (!fRoot)
          return -1;
       else
          return fRoot->FindRank(obj);
    }

    //______________________________________________________________________________
    TObject *TBtree::Remove(TObject *obj)
    {
       // Remove an object from the tree.

       if (!obj->IsSortable()) {
          Error("Remove", "object must be sortable");
          return 0;
       }
       if (!fRoot)
          return 0;

       TBtNode *loc;
       Int_t idx;
       TObject *ob = fRoot->Found(obj, &loc, &idx);
       if (!ob)
          return 0;
       loc->Remove(idx);
       return ob;
    }

    //______________________________________________________________________________
    void TBtree::RootIsFull()
    {
       // The root of the tree is full. Create an InnerNode that
       // points to it, and then inform the InnerNode that it is full.

       TBtNode *oldroot = fRoot;
       fRoot = new TBtInnerNode(0, this, oldroot);
       Assert(fRoot != 0);
       oldroot->Split();
    }

    //______________________________________________________________________________
    void TBtree::RootIsEmpty()
    {
       // If root is empty clean up its space.

       if (fRoot->fIsLeaf) {
          TBtLeafNode *lroot = (TBtLeafNode*)fRoot;
          Check(lroot->Psize() == 0);
          delete lroot;
          fRoot = 0;
       } else {
          TBtInnerNode *iroot = (TBtInnerNode*)fRoot;
          Check(iroot->Psize() == 0);
          fRoot = iroot->GetTree(0);
          fRoot->fParent = 0;
          delete iroot;
       }
    }

    //_______________________________________________________________________
    void TBtree::Streamer(TBuffer &b)
    {
       // Stream all objects in the btree to or from the I/O buffer.

       UInt_t R__s, R__c;
       if (b.IsReading()) {
          b.ReadVersion(&R__s, &R__c);   //Version_t v = b.ReadVersion();
          b >> fOrder;
          b >> fOrder2;
          b >> fInnerLowWaterMark;
          b >> fLeafLowWaterMark;
          b >> fInnerMaxIndex;
          b >> fLeafMaxIndex;
          TSeqCollection::Streamer(b);
          b.CheckByteCount(R__s, R__c,TBtree::IsA());
       } else {
          R__c = b.WriteVersion(TBtree::IsA(), kTRUE);
          b << fOrder;
          b << fOrder2;
          b << fInnerLowWaterMark;
          b << fLeafLowWaterMark;
          b << fInnerMaxIndex;
          b << fLeafMaxIndex;
          TSeqCollection::Streamer(b);
          b.SetByteCount(R__c, kTRUE);
       }
    }

    //////////////////////////////////////////////////////////////////////////
    //                                                                      //
    // TBtItem                                                              //
    //                                                                      //
    // Item stored in inner nodes of a TBtree.                              //
    //                                                                      //
    //////////////////////////////////////////////////////////////////////////

    //______________________________________________________________________________
    TBtItem::TBtItem()
    {
       // Create an item to be stored in the tree. An item contains a counter
       // of the number of keys (i.e. objects) in the node. A pointer to the
       // node and a pointer to the object being stored.

       fNofKeysInTree = 0;
       fTree = 0;
       fKey  = 0;
    }

    //______________________________________________________________________________
    TBtItem::TBtItem(TBtNode *n, TObject *obj)
    {
       // Create an item to be stored in the tree. An item contains a counter
       // of the number of keys (i.e. objects) in the node. A pointer to the
       // node and a pointer to the object being stored.

       fNofKeysInTree = n->NofKeys()+1;
       fTree = n;
       fKey  = obj;
    }

    //______________________________________________________________________________
    TBtItem::TBtItem(TObject *obj, TBtNode *n)
    {
       // Create an item to be stored in the tree. An item contains a counter
       // of the number of keys (i.e. objects) in the node. A pointer to the
       // node and a pointer to the object being stored.

       fNofKeysInTree = n->NofKeys()+1;
       fTree = n;
       fKey  = obj;
    }

    //______________________________________________________________________________
    TBtItem::~TBtItem()
    {
       // Delete a tree item.
    }

    //////////////////////////////////////////////////////////////////////////
    //                                                                      //
    // TBtNode                                                              //
    //                                                                      //
    // Abstract base class (ABC) of a TBtree node.                          //
    //                                                                      //
    //////////////////////////////////////////////////////////////////////////

    //______________________________________________________________________________
    TBtNode::TBtNode(Int_t isleaf, TBtInnerNode *p, TBtree *t)
    {
       // Create a B-tree node.

       fLast   = -1;
       fIsLeaf = isleaf;
       fParent = p;
       if (p == 0) {
          Check(t != 0);
          fTree = t;
       } else
    #ifdef cxxbug
    //  BUG in the cxx compiler. cxx complains that it cannot access fTree
    //  from TBtInnerNode. To reproduce the cxx bug uncomment the following line
    //  and delete the line after.
    //    fTree = p->fTree;
          fTree = p->GetParentTree();
    #else
          fTree = p->fTree;
    #endif
    }

    //______________________________________________________________________________
    TBtNode::~TBtNode()
    {
       // Delete a B-tree node.
    }

    //////////////////////////////////////////////////////////////////////////
    //                                                                      //
    // TBtreeIter                                                           //
    //                                                                      //
    // Iterator of btree.                                                   //
    //                                                                      //
    //////////////////////////////////////////////////////////////////////////

    ClassImp(TBtreeIter)

    //______________________________________________________________________________
     TBtreeIter::TBtreeIter(const TBtree *t, Bool_t dir)
                : fTree(t), fCursor(0), fDirection(dir)
    {
       // Create a B-tree iterator.

       Reset();
    }

    //______________________________________________________________________________
     TBtreeIter::TBtreeIter(const TBtreeIter &iter) : TIterator(iter)
    {
       // Copy ctor.

       fTree      = iter.fTree;
       fCursor    = iter.fCursor;
       fDirection = iter.fDirection;
    }

    //______________________________________________________________________________
    TIterator &TBtreeIter::operator=(const TIterator &rhs)
    {
       // Overridden assignment operator.

       if (this != &rhs && rhs.IsA() == TBtreeIter::Class()) {
          const TBtreeIter &rhs1 = (const TBtreeIter &)rhs;
          fTree      = rhs1.fTree;
          fCursor    = rhs1.fCursor;
          fDirection = rhs1.fDirection;
       }
       return *this;
    }

    //______________________________________________________________________________
    TBtreeIter &TBtreeIter::operator=(const TBtreeIter &rhs)
    {
       // Overloaded assignment operator.

       if (this != &rhs) {
          fTree      = rhs.fTree;
          fCursor    = rhs.fCursor;
          fDirection = rhs.fDirection;
       }
       return *this;
    }

    //______________________________________________________________________________
     void TBtreeIter::Reset()
    {
       // Reset the B-tree iterator.

       if (fDirection == kIterForward)
          fCursor = 0;
       else
          fCursor = fTree->GetSize() - 1;
    }

    //______________________________________________________________________________
     TObject *TBtreeIter::Next()
    {
       // Get next object from B-tree. Returns 0 when no more objects in tree.

       if (fDirection == kIterForward) {
          if (fCursor < fTree->GetSize())
             return (*fTree)[fCursor++];
       } else {
          if (fCursor >= 0)
             return (*fTree)[fCursor--];
       }
       return 0;
    }

    //////////////////////////////////////////////////////////////////////////
    //                                                                      //
    // TBtInnerNode                                                         //
    //                                                                      //
    // Inner node of a TBtree.                                              //
    //                                                                      //
    //////////////////////////////////////////////////////////////////////////

    //______________________________________________________________________________
    TBtInnerNode::TBtInnerNode(TBtInnerNode *p, TBtree *t) : TBtNode(0,p,t)
    {
       // Create a B-tree innernode.

       const Int_t index = MaxIndex() + 1;
       fItem = new TBtItem[ index ];
       if (fItem == 0)
          ::Fatal("TBtInnerNode::TBtInnerNode", "no more memory");
    }

    //______________________________________________________________________________
    TBtInnerNode::TBtInnerNode(TBtInnerNode *parent, TBtree *tree, TBtNode *oldroot)
                    : TBtNode(0, parent, tree)
    {
       // Called only by TBtree to initialize the TBtInnerNode that is
       // about to become the root.

       fItem = new TBtItem[MaxIndex()+1];
       if (fItem == 0)
          ::Fatal("TBtInnerNode::TBtInnerNode", "no more memory");
       Append(0, oldroot);
    }

    //______________________________________________________________________________
    TBtInnerNode::~TBtInnerNode()
    {
       if (fLast > 0)
          delete fItem[0].fTree;
       for (Int_t i = 1; i <= fLast; i++)
          delete fItem[i].fTree;

       delete [] fItem;
    }

    //______________________________________________________________________________
    void TBtInnerNode::Add(const TObject *obj, Int_t index)
    {
       // This is called only from TBtree::Add().

       Assert(index >= 1 && obj->IsSortable());
       TBtLeafNode *ln = GetTree(index-1)->LastLeafNode();
       ln->Add(obj, ln->fLast+1);
    }

    //______________________________________________________________________________
    void TBtInnerNode::AddElt(TBtItem &itm, Int_t at)
    {
       Assert(0 <= at && at <= fLast+1);
       Assert(fLast < MaxIndex());
       for (Int_t i = fLast+1; i > at ; i--)
          GetItem(i) = GetItem(i-1);
       SetItem(at, itm);
       fLast++;
    }

    //______________________________________________________________________________
    void TBtInnerNode::AddElt(Int_t at, TObject *k, TBtNode *t)
    {
       TBtItem newitem(k, t);
       AddElt(newitem, at);
    }

    //______________________________________________________________________________
    void TBtInnerNode::Add(TBtItem &itm, Int_t at)
    {
       AddElt(itm, at);
       if (IsFull())
          InformParent();
    }

    //______________________________________________________________________________
    void TBtInnerNode::Add(Int_t at, TObject *k, TBtNode *t)
    {
       TBtItem newitem(k, t);
       Add(newitem, at);
    }

    //______________________________________________________________________________
    void TBtInnerNode::AppendFrom(TBtInnerNode *src, Int_t start, Int_t stop)
    {
       // This should never create a full node that is, it is not used
       // anywhere where THIS could possibly be near full.

       if (start > stop)
          return;
       Assert(0 <= start && start <= src->fLast);
       Assert(0 <= stop  && stop  <= src->fLast );
       Assert(fLast + stop - start + 1 < MaxIndex()); // full-node check
       for (Int_t i = start; i <= stop; i++)
          SetItem(++fLast, src->GetItem(i));
    }

    //______________________________________________________________________________
    void TBtInnerNode::Append(TObject *d, TBtNode *n)
    {
       // Never called from anywhere where it might fill up THIS.

       Assert(fLast < MaxIndex());
       if (d) Assert(d->IsSortable());
       SetItem(++fLast, d, n);
    }

    //______________________________________________________________________________
    void TBtInnerNode::Append(TBtItem &itm)
    {
       Assert(fLast < MaxIndex());
       SetItem(++fLast, itm);
    }

    //______________________________________________________________________________
    void TBtInnerNode::BalanceWithLeft(TBtInnerNode *leftsib, Int_t pidx)
    {
       // THIS has more than LEFTSIB. Move some item from THIS to LEFTSIB.
       // PIDX is the index of the parent item that will change when keys
       // are moved.

       Assert(Vsize() >= leftsib->Psize());
       Assert(fParent->GetTree(pidx) == this);
       Int_t newThisSize = (Vsize() + leftsib->Psize())/2;
       Int_t noFromThis = Psize() - newThisSize;
       PushLeft(noFromThis, leftsib, pidx);
    }

    //______________________________________________________________________________
    void TBtInnerNode::BalanceWithRight(TBtInnerNode *rightsib, Int_t pidx)
    {
       // THIS has more than RIGHTSIB. Move some items from THIS to RIGHTSIB.
       // PIDX is the index of the parent item that will change when keys
       // are moved.

       Assert(Psize() >= rightsib->Vsize());
       Assert(fParent->GetTree(pidx) == rightsib);
       Int_t newThisSize = (Psize() + rightsib->Vsize())/2;
       Int_t noFromThis = Psize() - newThisSize;
       PushRight(noFromThis, rightsib, pidx);
    }

    //______________________________________________________________________________
    void TBtInnerNode::BalanceWith(TBtInnerNode *rightsib, Int_t pindx)
    {
       // PINDX is the index of the parent item whose key will change when
       // keys are shifted from one InnerNode to the other.

       if (Psize() < rightsib->Vsize())
          rightsib->BalanceWithLeft(this, pindx);
       else
          BalanceWithRight(rightsib, pindx);
    }

    //______________________________________________________________________________
    void TBtInnerNode::DecrNofKeys(TBtNode *that)
    {
       // THAT is a child of THIS that has just shrunk by 1.

       Int_t i = IndexOf(that);
       fItem[i].fNofKeysInTree--;
       if (fParent != 0)
          fParent->DecrNofKeys(this);
       else
          fTree->DecrNofKeys();
    }

    //______________________________________________________________________________
    Int_t TBtInnerNode::FindRank(const TObject *what) const
    {
       // Recursively look for WHAT starting in the current node.

       if (((TObject *)what)->Compare(GetKey(1)) < 0)
          return GetTree(0)->FindRank(what);
       Int_t sum = GetNofKeys(0);
       for (Int_t i = 1; i < fLast; i++) {
          if (((TObject*)what)->Compare(GetKey(i)) == 0)
             return sum;
          sum++;
          if (((TObject *)what)->Compare(GetKey(i+1)) < 0)
             return sum + GetTree(i)->FindRank(what);
          sum += GetNofKeys(i);
       }
       if (((TObject*)what)->Compare(GetKey(fLast)) == 0)
          return sum;
       sum++;
       // *what > GetKey(fLast), so recurse on last fItem.fTree
       return sum + GetTree(fLast)->FindRank(what);
    }

    //______________________________________________________________________________
    Int_t TBtInnerNode::FindRankUp(const TBtNode *that) const
    {
       // FindRankUp is FindRank in reverse.
       // Whereas FindRank looks for the object and computes the rank
       // along the way while walking DOWN the tree, FindRankUp already
       // knows where the object is and has to walk UP the tree from the
       // object to compute the rank.

       Int_t l   = IndexOf(that);
       Int_t sum = 0;
       for (Int_t i = 0; i < l; i++)
          sum += GetNofKeys(i);
       return sum + l + (fParent == 0 ? 0 : fParent->FindRankUp(this));
    }

    //______________________________________________________________________________
    TBtLeafNode *TBtInnerNode::FirstLeafNode()
    {
       return GetTree(0)->FirstLeafNode();
    }

    //______________________________________________________________________________
    TObject *TBtInnerNode::Found(const TObject *what, TBtNode **which, Int_t *where)
    {
       // Recursively look for WHAT starting in the current node.

       Assert(what->IsSortable());
       for (Int_t i = 1 ; i <= fLast; i++) {
          if (GetKey(i)->Compare(what) == 0) {
             // then could go in either fItem[i].fTree or fItem[i-1].fTree
             // should go in one with the most room, but that's kinda
             // hard to calculate, so we'll stick it in fItem[i].fTree
             *which = this;
             *where = i;
             return GetKey(i);
          }
          if (GetKey(i)->Compare(what) > 0)
             return GetTree(i-1)->Found(what, which, where);
       }
       // *what > *(*this)[fLast].fKey, so recurse on last fItem.fTree
       return GetTree(fLast)->Found(what, which, where);
    }

    //______________________________________________________________________________
    void TBtInnerNode::IncrNofKeys(TBtNode *that)
    {
       // THAT is a child of THIS that has just grown by 1.

       Int_t i = IndexOf(that);
       fItem[i].fNofKeysInTree++;
       if (fParent != 0)
          fParent->IncrNofKeys(this);
       else
          fTree->IncrNofKeys();
    }

    //______________________________________________________________________________
    Int_t TBtInnerNode::IndexOf(const TBtNode *that) const
    {
       // Returns a number in the range 0 to this->fLast
       // 0 is returned if THAT == fTree[0].

       for (Int_t i = 0; i <= fLast; i++)
          if (GetTree(i) == that)
             return i;
       Check(0);
       return 0;
    }

    //______________________________________________________________________________
    void TBtInnerNode::InformParent()
    {
       if (fParent == 0) {
          // then this is the root of the tree and needs to be split
          // inform the btree.
          Assert(fTree->fRoot == this);
          fTree->RootIsFull();
       } else
          fParent->IsFull(this);
    }

    //______________________________________________________________________________
    void TBtInnerNode::IsFull(TBtNode *that)
    {
       // The child node THAT is full. We will either redistribute elements
       // or create a new node and then redistribute.
       // In an attempt to minimize the number of splits, we adopt the following
       // strategy:
       //  * redistribute if possible
       //  * if not possible, then split with a sibling

       if (that->fIsLeaf) {
          TBtLeafNode *leaf = (TBtLeafNode *)that;
          TBtLeafNode *left = 0;
          TBtLeafNode *right= 0;
          // split LEAF only if both sibling nodes are full.
          Int_t leafidx = IndexOf(leaf);
          Int_t hasRightSib = (leafidx < fLast) &&
                               ((right = (TBtLeafNode*)GetTree(leafidx+1)) != 0);
          Int_t hasLeftSib  = (leafidx > 0) &&
                               ((left = (TBtLeafNode*)GetTree(leafidx-1)) != 0);
          Int_t rightSibFull = (hasRightSib && right->IsAlmostFull());
          Int_t leftSibFull  = (hasLeftSib  && left->IsAlmostFull());
          if (rightSibFull) {
             if (leftSibFull) {
                // both full, so pick one to split with
                left->SplitWith(leaf, leafidx);
             } else if (hasLeftSib) {
                // left sib not full, so balance with it
                leaf->BalanceWithLeft(left, leafidx);
             } else {
                // there is no left sibling, so split with right
                leaf->SplitWith(right, leafidx+1);
             }
          } else if (hasRightSib) {
             // right sib not full, so balance with it
             leaf->BalanceWithRight(right, leafidx+1);
          } else if (leftSibFull) {
             // no right sib, and left sib is full, so split with it
             left->SplitWith(leaf, leafidx);
          } else if (hasLeftSib) {
             // left sib not full so balance with it
             leaf->BalanceWithLeft(left, leafidx);
          } else {
             // neither a left or right sib; should never happen
             Check(0);
          }
       } else {
          TBtInnerNode *inner = (TBtInnerNode *)that;
          // split INNER only if both sibling nodes are full
          Int_t inneridx = IndexOf(inner);
          TBtInnerNode *left = 0;
          TBtInnerNode *right= 0;
          Int_t hasRightSib = (inneridx < fLast) &&
                               ((right = (TBtInnerNode*)GetTree(inneridx+1)) != 0);
          Int_t hasLeftSib  = (inneridx > 0) &&
                               ((left=(TBtInnerNode*)GetTree(inneridx-1)) != 0);
          Int_t rightSibFull = (hasRightSib && right->IsAlmostFull());
          Int_t leftSibFull  = (hasLeftSib  && left->IsAlmostFull());
          if (rightSibFull) {
             if (leftSibFull) {
                left->SplitWith(inner, inneridx);
             } else if (hasLeftSib) {
                inner->BalanceWithLeft(left, inneridx);
             } else {
                // there is no left sibling
                inner->SplitWith(right, inneridx+1);
             }
          } else if (hasRightSib) {
             inner->BalanceWithRight(right, inneridx+1);
          } else if (leftSibFull) {
             left->SplitWith(inner, inneridx);
          } else if (hasLeftSib) {
             inner->BalanceWithLeft(left, inneridx);
          } else {
             Check(0);
          }
       }
    }

    //______________________________________________________________________________
    void TBtInnerNode::IsLow(TBtNode *that)
    {
       // The child node THAT is <= half full. We will either redistribute
       // elements between children, or THAT will be merged with another child.
       // In an attempt to minimize the number of mergers, we adopt the following
       // strategy:
       //  * redistribute if possible
       //  * if not possible, then merge with a sibling

       if (that->fIsLeaf) {
          TBtLeafNode *leaf = (TBtLeafNode *)that;
          TBtLeafNode *left = 0;
          TBtLeafNode *right= 0;
          // split LEAF only if both sibling nodes are full.
          Int_t leafidx = IndexOf(leaf);
          Int_t hasRightSib = (leafidx < fLast) &&
                               ((right = (TBtLeafNode*)GetTree(leafidx+1)) != 0);
          Int_t hasLeftSib  = (leafidx > 0) &&
                               ((left = (TBtLeafNode*)GetTree(leafidx-1)) != 0);
          if (hasRightSib && (leaf->Psize() + right->Vsize()) >= leaf->MaxPsize()) {
             // then cannot merge,
             // and balancing this and rightsib will leave them both
             // more than half full
             leaf->BalanceWith(right, leafidx+1);
          } else if (hasLeftSib && (leaf->Vsize() + left->Psize()) >= leaf->MaxPsize()) {
             // ditto
             left->BalanceWith(leaf, leafidx);
          } else if (hasLeftSib) {
             // then they should be merged
             left->MergeWithRight(leaf, leafidx);
          } else if (hasRightSib) {
             leaf->MergeWithRight(right, leafidx+1);
          } else {
             Check(0); // should never happen
          }
       } else {
          TBtInnerNode *inner = (TBtInnerNode *)that;

          Int_t inneridx = IndexOf(inner);
          TBtInnerNode *left = 0;
          TBtInnerNode *right= 0;
          Int_t hasRightSib = (inneridx < fLast) &&
                               ((right = (TBtInnerNode*)GetTree(inneridx+1)) != 0);
          Int_t hasLeftSib  = (inneridx > 0) &&
                               ((left = (TBtInnerNode*)GetTree(inneridx-1)) != 0);
          if (hasRightSib && (inner->Psize() + right->Vsize()) >= inner->MaxPsize()) {
             // cannot merge
             inner->BalanceWith(right, inneridx+1);
          } else if (hasLeftSib && (inner->Vsize() + left->Psize()) >= inner->MaxPsize()) {
             // cannot merge
             left->BalanceWith(inner, inneridx);
          } else if (hasLeftSib) {
             left->MergeWithRight(inner, inneridx);
          } else if (hasRightSib) {
             inner->MergeWithRight(right, inneridx+1);
          } else {
             Check(0);
          }
       }
    }

    //______________________________________________________________________________
    TBtLeafNode *TBtInnerNode::LastLeafNode()
    {
        return GetTree(fLast)->LastLeafNode();
    }

    //______________________________________________________________________________
    void TBtInnerNode::MergeWithRight(TBtInnerNode *rightsib, Int_t pidx)
    {
       Assert(Psize() + rightsib->Vsize() < MaxIndex());
       if (rightsib->Psize() > 0)
          rightsib->PushLeft(rightsib->Psize(), this, pidx);
       rightsib->SetKey(0, fParent->GetKey(pidx));
       AppendFrom(rightsib, 0, 0);
       fParent->IncNofKeys(pidx-1, rightsib->GetNofKeys(0)+1);
       fParent->RemoveItem(pidx);
       delete rightsib;
    }

    //______________________________________________________________________________
    Int_t TBtInnerNode::NofKeys() const
    {
       Int_t sum = 0;
       for (Int_t i = 0; i <= fLast; i++)
          sum += GetNofKeys(i);
       return sum + Psize();
    }

    //______________________________________________________________________________
    TObject *TBtInnerNode::operator[](Int_t idx) const
    {
       for (Int_t j = 0; j <= fLast; j++) {
          Int_t r;
          if (idx < (r = GetNofKeys(j)))
             return (*GetTree(j))[idx];
          if (idx == r) {
             if (j == fLast) {
                ::Error("TBtInnerNode::operator[]", "should not happen, 0 returned");
                return 0;
              } else
                return GetKey(j+1);
          }
          idx -= r+1; // +1 because of the key in the node
       }
       ::Error("TBtInnerNode::operator[]", "should not happen, 0 returned");
       return 0;
    }

    //______________________________________________________________________________
    void TBtInnerNode::PushLeft(Int_t noFromThis, TBtInnerNode *leftsib, Int_t pidx)
    {
       // noFromThis==1 => moves the parent item into the leftsib,
       // and the first item in this's array into the parent item.

       Assert(fParent->GetTree(pidx) == this);
       Assert(noFromThis > 0 && noFromThis <= Psize());
       Assert(noFromThis + leftsib->Psize() < MaxPsize());
       SetKey(0, fParent->GetKey(pidx)); // makes AppendFrom's job easier
       leftsib->AppendFrom(this, 0, noFromThis-1);
       ShiftLeft(noFromThis);
       fParent->SetKey(pidx, GetKey(0));
       fParent->SetNofKeys(pidx-1, leftsib->NofKeys());
       fParent->SetNofKeys(pidx, NofKeys());
    }

    //______________________________________________________________________________
    void TBtInnerNode::PushRight(Int_t noFromThis, TBtInnerNode *rightsib, Int_t pidx)
    {
       Assert(noFromThis > 0 && noFromThis <= Psize());
       Assert(noFromThis + rightsib->Psize() < rightsib->MaxPsize());
       Assert(fParent->GetTree(pidx) == rightsib);
       //
       // The operation is three steps:
       //  Step I.   Make room for the incoming keys in RIGHTSIB.
       //  Step II.  Move the items from THIS into RIGHTSIB.
       //  Step III. Update the length of THIS.
       //
       // Step I. Make space for noFromThis items
       //
       Int_t start = fLast - noFromThis + 1;
       Int_t tgt, src;
       tgt = rightsib->fLast + noFromThis;
       src = rightsib->fLast;
       rightsib->fLast = tgt;
       rightsib->SetKey(0, fParent->GetKey(pidx)); IncNofKeys(0);
       while (src >= 0) {
          // do this kind of assignment on TBtInnerNode items only when
          // the parent fields of the moved items do not change, as they
          // don't here.
          // Otherwise, use SetItem so the parents are updated appropriately.
          rightsib->GetItem(tgt--) = rightsib->GetItem(src--);
       }

       // Step II. Move the items from THIS into RIGHTSIB
       for (Int_t i = fLast; i >= start; i-- ) {
          // this is the kind of assignment to use when parents change
          rightsib->SetItem(tgt--, GetItem(i));
       }
       fParent->SetKey(pidx, rightsib->GetKey(0));
       DecNofKeys(0);
       Check(tgt == -1);

       // Step III.
       fLast -= noFromThis;

       // Step VI.  update NofKeys
       fParent->SetNofKeys(pidx-1, NofKeys());
       fParent->SetNofKeys(pidx, rightsib->NofKeys());
    }

    //______________________________________________________________________________
    void TBtInnerNode::Remove(Int_t index)
    {
       Assert(index >= 1 && index <= fLast);
       TBtLeafNode *lf = GetTree(index)->FirstLeafNode();
       SetKey(index, lf->fItem[0]);
       lf->RemoveItem(0);
    }

    //______________________________________________________________________________
    void TBtInnerNode::RemoveItem(Int_t index)
    {
       Assert(index >= 1 && index <= fLast);
       for (Int_t to = index; to < fLast; to++)
          fItem[to] = fItem[to+1];
       fLast--;
       if (IsLow()) {
          if (fParent == 0) {
             // then this is the root; when only one child, make the child the root
             if (Psize() == 0)
                fTree->RootIsEmpty();
          } else
             fParent->IsLow(this);
       }
    }

    //______________________________________________________________________________
    void TBtInnerNode::ShiftLeft(Int_t cnt)
    {
       if (cnt <= 0)
          return;
       for (Int_t i = cnt; i <= fLast; i++)
          GetItem(i-cnt) = GetItem(i);
       fLast -= cnt;
    }

    //______________________________________________________________________________
    void TBtInnerNode::Split()
    {
       // This function is called only when THIS is the only descendent
       // of the root node, and THIS needs to be split.
       // Assumes that idx of THIS in fParent is 0.

       TBtInnerNode *newnode = new TBtInnerNode(fParent);
       Check(newnode != 0);
       fParent->Append(GetKey(fLast), newnode);
       newnode->AppendFrom(this, fLast, fLast);
       fLast--;
       fParent->IncNofKeys(1, newnode->GetNofKeys(0));
       fParent->DecNofKeys(0, newnode->GetNofKeys(0));
       BalanceWithRight(newnode, 1);
    }

    //______________________________________________________________________________
    void TBtInnerNode::SplitWith(TBtInnerNode *rightsib, Int_t keyidx)
    {
       // THIS and SIB are too full; create a NEWNODE, and balance
       // the number of keys between the three of them.
       //
       // picture: (also see Knuth Vol 3 pg 478)
       //               keyidx keyidx+1
       //            +--+--+--+--+--+--...
       //            |  |  |  |  |  |
       // fParent--->|  |     |     |
       //            |  |     |     |
       //            +*-+*-+*-+--+--+--...
       //             |  |  |
       //        +----+  |  +-----+
       //        |       +-----+  |
       //        V             |  V
       //        +----------+  |  +----------+
       //        |          |  |  |          |
       //  this->|          |  |  |          |<--sib
       //        +----------+  |  +----------+
       //                      V
       //                    data
       //
       // keyidx is the index of where the sibling is, and where the
       // newly created node will be recorded (sibling will be moved to
       // keyidx+1)

       Assert(keyidx > 0 && keyidx <= fParent->fLast);

       rightsib->SetKey(0, fParent->GetKey(keyidx));
       Int_t nofKeys      = Psize() + rightsib->Vsize();
       Int_t newSizeThis  = nofKeys / 3;
       Int_t newSizeNew   = (nofKeys - newSizeThis) / 2;
       Int_t newSizeSib   = (nofKeys - newSizeThis - newSizeNew);
       Int_t noFromThis   = Psize() - newSizeThis;
       Int_t noFromSib    = rightsib->Vsize() - newSizeSib;
       // because of their smaller size, this TBtInnerNode may not have to
       // give up any elements to the new node.  I.e., noFromThis == 0.
       // This will not happen for TBtLeafNodes.
       // We handle this by pulling an item from the rightsib.
       Check(noFromThis >= 0);
       Check(noFromSib >= 1);
       TBtInnerNode *newNode = new TBtInnerNode(fParent);
       Check(newNode != 0);
       if (noFromThis > 0) {
          newNode->Append(GetItem(fLast));
          fParent->AddElt(keyidx, GetKey(fLast--), newNode);
          if (noFromThis > 2)
             this->PushRight(noFromThis-1, newNode, keyidx);
          rightsib->PushLeft(noFromSib, newNode, keyidx+1);
       } else {
          // pull an element from the rightsib
          newNode->Append(rightsib->GetItem(0));
          fParent->AddElt(keyidx+1, rightsib->GetKey(1), rightsib);
          rightsib->ShiftLeft(1);
          fParent->SetTree(keyidx, newNode);
          rightsib->PushLeft(noFromSib-1, newNode, keyidx+1);
       }
       fParent->SetNofKeys(keyidx-1, this->NofKeys());
       fParent->SetNofKeys(keyidx, newNode->NofKeys());
       fParent->SetNofKeys(keyidx+1, rightsib->NofKeys());
       if (fParent->IsFull())
          fParent->InformParent();
    }

    //////////////////////////////////////////////////////////////////////////
    //                                                                      //
    // TBtLeafNode                                                          //
    //                                                                      //
    // Leaf node of a TBtree.                                               //
    //                                                                      //
    //////////////////////////////////////////////////////////////////////////

    //______________________________________________________________________________
    TBtLeafNode::TBtLeafNode(TBtInnerNode *p, const TObject *obj, TBtree *t): TBtNode(1, p, t)
    {
       fItem = new TObject *[MaxIndex()+1];
       memset(fItem, 0, (MaxIndex()+1)*sizeof(TObject*));

       Assert(fItem != 0);
       if (obj != 0)
          fItem[++fLast] = (TObject*)obj;   // cast const away
    }

    //______________________________________________________________________________
    TBtLeafNode::~TBtLeafNode()
    {
       delete [] fItem;
    }

    //______________________________________________________________________________
    void TBtLeafNode::Add(const TObject *obj, Int_t index)
    {
       // Add the object OBJ to the leaf node, inserting it at location INDEX
       // in the fItem array.

       Assert(obj->IsSortable());
       Assert(0 <= index && index <= fLast+1);
       Assert(fLast <= MaxIndex());
       for (Int_t i = fLast+1; i > index ; i--)
          fItem[i] = fItem[i-1];
       fItem[index] = (TObject *)obj;
       fLast++;

       // check for overflow
       if (fParent == 0)
          fTree->IncrNofKeys();
       else
          fParent->IncrNofKeys(this);

       if (IsFull()) {
          // it's full; tell parent node
          if (fParent == 0) {
             // this occurs when this leaf is the only node in the
             // btree, and this->fTree->fRoot == this
             Check(fTree->fRoot == this);
             // in which case we inform the btree, which can be
             // considered the parent of this node
             fTree->RootIsFull();
          } else {
             // the parent is responsible for splitting/balancing subnodes
             fParent->IsFull(this);
          }
       }
    }

    //______________________________________________________________________________
    void TBtLeafNode::AppendFrom(TBtLeafNode *src, Int_t start, Int_t stop)
    {
       // A convenience function, does not worry about the element in
       // the parent, simply moves elements from SRC[start] to SRC[stop]
       // into the current array.
       // This should never create a full node.
       // That is, it is not used anywhere where THIS could possibly be
       // near full.
       // Does NOT handle nofKeys.

       if (start > stop)
          return;
       Assert(0 <= start && start <= src->fLast);
       Assert(0 <= stop  && stop  <= src->fLast);
       Assert(fLast + stop - start + 1 < MaxIndex()); // full-node check
       for (Int_t i = start; i <= stop; i++)
          fItem[++fLast] = src->fItem[i];
       Check(fLast < MaxIndex());
    }

    //______________________________________________________________________________
    void TBtLeafNode::Append(TObject *obj)
    {
       // Never called from anywhere where it might fill up THIS
       // does NOT handle nofKeys.

       Assert(obj->IsSortable());
       fItem[++fLast] = obj;
       Check(fLast < MaxIndex());
    }

    //______________________________________________________________________________
    void TBtLeafNode::BalanceWithLeft(TBtLeafNode *leftsib, Int_t pidx)
    {
       // THIS has more than LEFTSIB;  move some items from THIS to LEFTSIB.

       Assert(Vsize() >= leftsib->Psize());
       Int_t newThisSize = (Vsize() + leftsib->Psize())/2;
       Int_t noFromThis  = Psize() - newThisSize;
       PushLeft(noFromThis, leftsib, pidx);
    }

    //______________________________________________________________________________
    void TBtLeafNode::BalanceWithRight(TBtLeafNode *rightsib, Int_t pidx)
    {
       // THIS has more than RIGHTSIB;  move some items from THIS to RIGHTSIB.

       Assert(Psize() >= rightsib->Vsize());
       Int_t newThisSize = (Psize() + rightsib->Vsize())/2;
       Int_t noFromThis  = Psize() - newThisSize;
       PushRight(noFromThis, rightsib, pidx);
    }

    //______________________________________________________________________________
    void TBtLeafNode::BalanceWith(TBtLeafNode *rightsib, Int_t pidx)
    {
       // PITEM is the parent item whose key will change when keys are shifted
       // from one LeafNode to the other.

       if (Psize() < rightsib->Vsize())
          rightsib->BalanceWithLeft(this, pidx);
       else
          BalanceWithRight(rightsib, pidx);
    }

    //______________________________________________________________________________
    Int_t TBtLeafNode::FindRank(const TObject *what) const
    {
       // WHAT was not in any inner node; it is either here, or it's
       // not in the tree.

       for (Int_t i = 0; i <= fLast; i++) {
          if (fItem[i]->Compare(what) == 0)
             return i;
          if (fItem[i]->Compare(what) > 0)
             return -1;
       }
       return -1;
    }

    //______________________________________________________________________________
    TBtLeafNode *TBtLeafNode::FirstLeafNode()
    {
       return this;
    }

    //______________________________________________________________________________
    TObject *TBtLeafNode::Found(const TObject *what, TBtNode **which, Int_t *where)
    {
       // WHAT was not in any inner node; it is either here, or it's
       // not in the tree.

       Assert(what->IsSortable());
       for (Int_t i = 0; i <= fLast; i++) {
          if (fItem[i]->Compare(what) == 0) {
             *which = this;
             *where = i;
             return fItem[i];
          }
          if (fItem[i]->Compare(what) > 0) {
             *which = this;
             *where = i;
             return 0;
          }
       }
       *which = this;
       *where = fLast+1;
       return 0;
    }

    //______________________________________________________________________________
    Int_t TBtLeafNode::IndexOf(const TObject *that) const
    {
       // Returns a number in the range 0 to MaxIndex().

       for (Int_t i = 0; i <= fLast; i++) {
          if (fItem[i] == that)
             return i;
       }
       Check(0);
       return -1;
    }

    //______________________________________________________________________________
    TBtLeafNode *TBtLeafNode::LastLeafNode()
    {
       return this;
    }

    //______________________________________________________________________________
    void TBtLeafNode::MergeWithRight(TBtLeafNode *rightsib, Int_t pidx)
    {
       Assert(Psize() + rightsib->Vsize() < MaxPsize());
       rightsib->PushLeft(rightsib->Psize(), this, pidx);
       Append(fParent->GetKey(pidx));
       fParent->SetNofKeys(pidx-1, NofKeys());
       fParent->RemoveItem(pidx);
       delete rightsib;
    }

    //______________________________________________________________________________
    Int_t TBtLeafNode::NofKeys(Int_t ) const
    {
       return 1;
    }

    //______________________________________________________________________________
    Int_t TBtLeafNode::NofKeys() const
    {
       return Psize();
    }

    //______________________________________________________________________________
    //void TBtLeafNode::PrintOn(ostream& out) const
    //{
    //    out << " < ";
    //    for (Int_t i = 0; i <= fLast; i++)
    //        out << *fItem[i] << " " ;
    //    out << "> ";
    //}

    //______________________________________________________________________________
    void TBtLeafNode::PushLeft(Int_t noFromThis, TBtLeafNode *leftsib, Int_t pidx)
    {
       // noFromThis==1 => moves the parent item into the leftsib,
       // and the first item in this's array into the parent item.

       Assert(noFromThis > 0 && noFromThis <= Psize());
       Assert(noFromThis + leftsib->Psize() < MaxPsize());
       Assert(fParent->GetTree(pidx) == this);
       leftsib->Append(fParent->GetKey(pidx));
       if (noFromThis > 1)
          leftsib->AppendFrom(this, 0, noFromThis-2);
       fParent->SetKey(pidx, fItem[noFromThis-1]);
       ShiftLeft(noFromThis);
       fParent->SetNofKeys(pidx-1, leftsib->NofKeys());
       fParent->SetNofKeys(pidx, NofKeys());
    }

    //______________________________________________________________________________
    void TBtLeafNode::PushRight(Int_t noFromThis, TBtLeafNode *rightsib, Int_t pidx)
    {
       // noFromThis==1 => moves the parent item into the
       // rightsib, and the last item in this's array into the parent
       // item.

       Assert(noFromThis > 0 && noFromThis <= Psize());
       Assert(noFromThis + rightsib->Psize() < MaxPsize());
       Assert(fParent->GetTree(pidx) == rightsib);
       // The operation is five steps:
       //  Step I.   Make room for the incoming keys in RIGHTSIB.
       //  Step II.  Move the key in the parent into RIGHTSIB.
       //  Step III. Move the items from THIS into RIGHTSIB.
       //  Step IV.  Move the item from THIS into the parent.
       //  Step V.   Update the length of THIS.
       //
       // Step I.: make space for noFromThis items
       //
       Int_t start = fLast - noFromThis + 1;
       Int_t tgt, src;
       tgt = rightsib->fLast + noFromThis;
       src = rightsib->fLast;
       rightsib->fLast = tgt;
       while (src >= 0)
          rightsib->fItem[tgt--] = rightsib->fItem[src--];

       // Step II. Move the key from the parent into place
       rightsib->fItem[tgt--] = fParent->GetKey(pidx);

       // Step III.Move the items from THIS into RIGHTSIB
       for (Int_t i = fLast; i > start; i--)
          rightsib->fItem[tgt--] = fItem[i];
       Check(tgt == -1);

       // Step IV.
       fParent->SetKey(pidx, fItem[start]);

       // Step V.
       fLast -= noFromThis;

       // Step VI.  update nofKeys
       fParent->SetNofKeys(pidx-1, NofKeys());
       fParent->SetNofKeys(pidx, rightsib->NofKeys());
    }

    //______________________________________________________________________________
    void TBtLeafNode::Remove(Int_t index)
    {
       Assert(index >= 0 && index <= fLast);
       for (Int_t to = index; to < fLast; to++)
          fItem[to] = fItem[to+1];
       fLast--;
       if (fParent == 0)
          fTree->DecrNofKeys();
       else
          fParent->DecrNofKeys(this);
       if (IsLow()) {
          if (fParent == 0) {
             // then this is the root; when no keys left, inform the tree
             if (Psize() == 0)
                fTree->RootIsEmpty();
          } else
             fParent->IsLow(this);
       }
    }

    //______________________________________________________________________________
    void TBtLeafNode::ShiftLeft(Int_t cnt)
    {
       if (cnt <= 0)
          return;
       for (Int_t i = cnt; i <= fLast; i++)
          fItem[i-cnt] = fItem[i];
       fLast -= cnt;
    }

    //______________________________________________________________________________
    void TBtLeafNode::Split()
    {
       // This function is called only when THIS is the only descendent
       // of the root node, and THIS needs to be split.
       // Assumes that idx of THIS in Parent is 0.

       TBtLeafNode *newnode = new TBtLeafNode(fParent);
       Assert(newnode != 0);
       fParent->Append(fItem[fLast--], newnode);
       fParent->SetNofKeys(0, fParent->GetTree(0)->NofKeys());
       fParent->SetNofKeys(1, fParent->GetTree(1)->NofKeys());
       BalanceWithRight(newnode, 1);
    }

    //______________________________________________________________________________
    void TBtLeafNode::SplitWith(TBtLeafNode *rightsib, Int_t keyidx)
    {
       Assert(fParent == rightsib->fParent);
       Assert(keyidx > 0 && keyidx <= fParent->fLast);
       Int_t nofKeys      = Psize() + rightsib->Vsize();
       Int_t newSizeThis  = nofKeys / 3;
       Int_t newSizeNew   = (nofKeys - newSizeThis) / 2;
       Int_t newSizeSib   = (nofKeys - newSizeThis - newSizeNew);
       Int_t noFromThis   = Psize() - newSizeThis;
       Int_t noFromSib    = rightsib->Vsize() - newSizeSib;
       Check(noFromThis >= 0);
       Check(noFromSib >= 1);
       TBtLeafNode *newNode  = new TBtLeafNode(fParent);
       Assert(newNode != 0);
       fParent->AddElt(keyidx, fItem[fLast--], newNode);
       fParent->SetNofKeys(keyidx, 0);
       fParent->DecNofKeys(keyidx-1);
       this->PushRight(noFromThis-1, newNode, keyidx);
       rightsib->PushLeft(noFromSib, newNode, keyidx+1);
       if (fParent->IsFull())
          fParent->InformParent();
    }