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Posted 22 Dec 2004

# Fast Binary Tree Operations

, 22 Jan 2005
Describes main binary tree operations.

## Introduction

In this article, I will show you how to use the Binary Search Tree to store data. And I am using templates for keys and data values to simplify usage of any data types. If you have good experience about the theory behind the binary search trees, you can skip next sections of the background.

## Background

[1] Binary search trees are data structures which support many dynamic-set operations including search, insert, delete, minimum, maximum, predecessor, and successor. Basic operations on binary search tree take time proportional to the height of the tree. For a complete binary tree with n nodes, such operations take O(log n) time in the worst-case, as the height of randomly built binary search tree is proportional to log n. In some cases operations take O(n) time in case of sorted input, so the tree will be like a sorted list, so I used the Red-Black tree balancing technique, as described in the Tree Load Balancing section

A binary search tree is organized in a binary tree as shown in figure 1. Such a tree can be represented by a linked data structure in which each node is an object. In addition to the key field, each node contains fields left, right, and parent that point to the nodes corresponding to its left child, its right child, and its parent, respectively. If a child or the parent is missing, the appropriate field field contains a value NIL.

I will not dig in all Binary Tree functions here, I will just refer to the search function, and you can check reference [1] for more details, and I will talk about some of them in the code description, as many of them is so sophisticated like the Delete function.

#### Searching

The most common operation performed in binary search tree is searching for a key stored in the tree, which can be done in two ways:

1. Recursion:

Node Search(Node, key)
{
if Node = NIL or key = Node.key
return Node
if key < Node.key
then return Search(Node.Left, key)
else return Search(Node.Right, key)
}
2. While loop:
Node Search(key)
{
Node = root
while Node != NIL and key != Node.key
do if key < Node.key
then Node = Node.Left
else Node = Node.Right
return Node
}

The two ways begin search at the root and trace a path downward in the tree till they find the key or return NIL (not found case). First way code is more simple, but second way optimizes stack usage, which I am preferring in all my cases to deal with huge data without affecting the stack and the performance.

## Code Description

All Tree functions encapsulated in a template class CBinaryTree and CBinaryTreeNode.

 GetCount returns tree nodes count with repetition (repeated key will assigned to one node). RemoveAll removes all tree nodes. Insert inserts new key in the tree. Search searches for key in the tree. Min gets minimum node key under the input node. Max gets maximum node key under the input node. Successor gets node successor (node which comes next in the over all tree sorting). Predecessor gets node predecessor (node which comes previous in the over all tree sorting). Delete deletes node from tree and adjusts its childs' nodes. Save saves all tree nodes' order in a vector of integers.

In all my code here, I am avoiding recursion functions to avoid stack overflow, as I am dealing with huge data, so you will find all my code using <CODE lang=c++>while loops, sometimes code becomes complicated like the RemoveAll function:

#### RemoveAll function

This function removes all tree nodes, it does that by order to each node to delete its left child then its right child, then delete itself. This can be done with two ways:

1. While loop:
// remove all tree nodes
void RemoveAll()
{
TREENODE *node = Root, *pTemp;
while(node != Nil)
{
// check for left child
if(node->Left != Nil)
node = node->Left;
// check for right child
else    if(node->Right != Nil)
node = node->Right;
else    // node has no childs
{    // save node pointer
pTemp = node;
// set node pointer at its parent to NULL
if(node->Parent != Nil)
node->Parent->Childs[node != node->Parent->Left] = Nil;
// update pointer node to its parent
node = node->Parent;
// delete the saved node
delete pTemp;
}
}
Count = Serial = 0;
Root = Nil;
Modified = false;
}

As you can see, there is some complication at this section (node has no children).

2. Recursion way:

It simply puts delete left and right nodes at each node destructor and call delete for tree root.

~CBinaryTreeNode()
{
if(Childs[0])
delete Childs[0];
if(Childs[1])
delete Childs[1];
}
// ...
delete Tree.Root;

All the nodes will be deleted automatically through stack usage, but if you read the comments in the first way code, you will find it easy to understand it, and avoid stack usage.

#### Min function

The Minimum of a node x can be found by following left child from the node x until a Nil is encountered, as in the following code:

// return minimum key in the tree
TREENODE* Min(TREENODE* node) const
{
// iterate in the left branch
while(node != Nil && node->Left != Nil)
node = node->Left;
return node;
}

#### Successor function

The successor of a node x is the node with the smallest key greater than key[x]. For example, if you add keys (C I D H B F E G A J K) in sequence and will be as shown in the following figure:

Each arrow represents the direction from each node to its successor. The code in the Successorfunction takes two paths:

1. If the node has right child then return the left most node in the right sub tree.
2. Else, it goes up from node until we find a node that is to the left of its parent.
TREENODE* Successor(TREENODE* node) const
{
// return the left most node in the right sub tree
if(node->Right != Nil)
return Min(node->Right);
// go up from node until we find a node that is the left of its parent
TREENODE* Parent = node->Parent;
while(Parent != Nil && node == Parent->Right)
{
node = Parent;
Parent = node->Parent;
}
return Parent;
}

You can use the Successor function with the Min function to iterate tree nodes in ascending order.

#### Delete function

Delete function has three cases, I found it too complicated, and I hope I can describe it, The three cases are:

1. The node has no child, so we just remove it. As in you can see in figure 2, node K can be deleted by just resetting the pointer from node J.
2. The node has one child, so we need to splice it as node H in figure 2, we should make node D point to node F as a right child, and set node F parent to point to node D
3. The node has two children, so we choose its successor to take its position, and splice its successor (join successor parent and child, like the previous case) . For example, if we delete node C at figure 2, we get node C successor (D), so we will splice node D so that node I and H will be connected directly and node C will take node C place as in Figure 3.

Code in the Delete is optimized to handle the three cases altogether, so I find it hard to describe it here, but you can try to apply the code in each case and read comments carefully, and you will find it working well.

void Delete(TREENODE* node)
{
TREENODE *pSplice = (node->Left == Nil ||
node->Right == Nil)?node:Successor(node);
TREENODE *pChild = pSplice->Childs[pSplice->Left == Nil];
// connect child to spliced node parent
if(pChild != Nil)
pChild->Parent = pSplice->Parent;
// connect spliced node parent to child
if(pSplice->Parent == Nil)
Root = pChild;
else
pSplice->Parent->Childs[pSplice !=
pSplice->Parent->Childs[0]] = pChild;
// put spliced node in place of node (if required)
if(pSplice != node)
{    // copy spliced node
*node = *pSplice;
// delete the spliced node
delete pSplice;
}
else
// delete the node
delete node;
Count--;
}

#### Iterate Tree Nodes

You can call Min(Tree.Root) and then Successor in a loop to retrieve all nodes in order (ascending):

TREENODE* node = Min(Root);
while(node)
{
// use node here then get next one
node = Successor(node);
}

And in the same way you can call Max(Tree.Root) and then Predecessor in a loop to retrieve all nodes in a reversed order (descending):

TREENODE* node = Max(Root);
while(node)
{
// use node here then get next one
node = Predecessor(node);
}

## Class Usage

CBinaryTree usage is very simple, we just need to decide the KEY and DATA data types, then you can define the class, but you must choose a KEY that supports the compare function as it is used in the Insert and Search functions. For example:

CBiaryTree<string, LPCSTR, int, int> tree;
tree.NoRepeat = true;
for(...)
{
// ... fill str in some way
tree.Insert(str);
}

class String of the STL supports the compare function, but in some cases you have to add it yourself, as in case of integers' sort:

class CInt
{
public:
int m_nKey;
int compare(int nKey)
{ return nKey == m_nKey ? 0 : (nKey >= m_nKey ? -1 : 1); }
void operator=(int nKey) { m_nKey = nKey; }
bool operator==(int nKey) { return m_nKey == nKey; }
};

CBiaryTree<CInt, int, int, int> tree;
tree.NoRepeat = true;
for(...)
{
// ... fill n in some way
tree.Insert(n);
}

Tree balancing can be achieved using many techniques. In my class here, I am using Red-Black Tree functions for the insertion in the tree. Red-Black Tree simply depends on keep tree height short as possible, as the search and the insertion operations time depend on the tree height. As in the following figure, if the sequence (C, A, B) is added to the tree, the height will be like case 1, but if we have an operation to change to the valid tree in case 3, it will be good as the tree height reduced.

So, I have included the functions LeftRotate, RightRotate, and RBInsert to can balance the tree functions.

## Sample Demo

The attached source zip file with article contains a sample that parse any folder in a recursive function and parse all its files with the extension specified in the extensions editor (any text files format), and it adds all files tokens to a binary tree. Then you can navigate all tree tokens through the tree control. You can test it in VC6 or 7.1, or just run the attached exe, and don't hesitate to mail me for any help.

## Source code files

1. BinaryTree.h: Binary Tree code.
2. RBTree.h: Red-Black Tree code.

## References

[1] Thomas H.Cormen, Charles E.Leiserson, Ronald L.Rivest, Algorithms 1990

## Thanks to...

I awe a lot to my colleagues for helping me in implementing and testing this code. (JAK)

A list of licenses authors might use can be found here

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