Binary Trees and Binary Search Trees

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Binary Trees andBinary Search Trees

Based on Dale & Co: Object-Oriented Data Structures using C++ (graphics)

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A binary tree is a structure in which:

Each node can have at most two children, and in which a unique path exists from the root to every other node.

The two children of a node are called the left child and the right child, if they exist.

Binary Tree

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A Binary Tree

Q

V

T

K S

A E

L

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How many leaf nodes?

Q

V

T

K S

A E

L

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How many descendants of Q?

Q

V

T

K S

A E

L

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How many ancestors of K?

Q

V

T

K S

A E

L

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A Binary Tree with Tree Nodes

Q

V

T

K S

A E

L

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Node Terminology for a Tree Node

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Binary Trees

You can write a class that represents each node in a binary tree Called BinaryTreeNode

Instance variables of the class BinaryTreeNode info: stores the information part of the node lLink: points to the root node of the left subtree rLink: points to the root node of the right subtree

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Tree Traversals

Insertion, deletion, and lookup (and other) operations require that the binary tree be traversed

Commonly used traversals Inorder traversal Preorder traversal Postorder traversal

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Tree Traversals: In Order Print

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Inorder (tree): LeftNodeRight Binary tree is traversed as follows:

Traverse left subtree Visit node Traverse right subtree

private void inorder(BinaryTreeNode<T> t){ if (t != null) { inorder(t.lLink); System.out.print(t.info + “ “); inorder(t.rLink); }}

To print in alphabetical order

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Postorder (tree): LeftRightNode Binary tree is traversed as follows:

Traverse left subtree Traverse right subtree Visit node

private void postorder(BinaryTreeNode<T> t){ if (t != null){ postorder(t.lLink); postorder(t.rLink); System.out.print(t.info + “ “); } } Visits leaves first (good for deletion)

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Preorder (tree): NodeLeftRight Binary tree is traversed as follows:

Visit node Traverse left subtree Traverse right subtree

private void preorder(BinaryTreeNode<T> t){ if (t != null) { System.out.print(t.info + “ “); preorder(t.lLink); preorder(t.rLink); }} Useful with binary trees (not BST)

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Three Tree Traversals

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Binary Search Trees (BST)

To search for an item in a normal binary tree, you must traverse entire tree until item is found Search process will be very slow Similar to searching in an arbitrary linked list

Binary search tree Data in each node is

– Larger than the data in its left child– Smaller than the data in its right child

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A special kind of binary tree in which:

1. Each node contains a distinct data value,

2. The key values in the tree can be compared using “greater than” and “less than”, and

3. The key value of each node in the tree is

less than every key value in its right subtree, and greater than every key value in its left subtree.

So, a Binary Search Tree (BST) is . . .

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Depends on its key values and their order of insertion.

Insert the elements ‘J’ ‘E’ ‘F’ ‘T’ ‘A’ in that order.

The first value to be inserted is put into the root node.

The shape of a binary search tree . . .

‘J’

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Thereafter, each value to be inserted begins by comparing itself to the value in the root node, moving left it is less, or moving right if it is greater. This continues at each level until it can be inserted as a new leaf.

Inserting ‘E’ into the BST

‘J’

‘E’

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Begin by comparing ‘F’ to the value in the root node, moving left it is less, or moving right if it is greater. This continues until it can be inserted as a leaf.

Inserting ‘F’ into the BST

‘J’

‘E’

‘F’

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Begin by comparing ‘T’ to the value in the root node, moving left it is less, or moving right if it is greater. This continues until it can be inserted as a leaf.

Inserting ‘T’ into the BST

‘J’

‘E’

‘F’

‘T’

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Begin by comparing ‘A’ to the value in the root node, moving left it is less, or moving right if it is greater. This continues until it can be inserted as a leaf.

Inserting ‘A’ into the BST

‘J’

‘E’

‘F’

‘T’

‘A’

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is obtained by inserting

the elements ‘A’ ‘E’ ‘F’ ‘J’ ‘T’ in that order?

What binary search tree . . .

‘A’

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obtained by inserting

the elements ‘A’ ‘E’ ‘F’ ‘J’ ‘T’ in that order.

Binary search tree . . .

‘A’

‘E’

‘F’

‘J’

‘T’

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Another binary search tree

Add nodes containing these values in this order:

‘D’ ‘B’ ‘L’ ‘Q’ ‘S’ ‘V’ ‘Z’

‘J’

‘E’

‘A’ ‘H’

‘T’

‘M’

‘K’ ‘P’

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Is ‘F’ in the binary search tree?

‘J’

‘E’

‘A’ ‘H’

‘T’

‘M’

‘K’

‘V’

‘P’ ‘Z’‘D’

‘Q’‘L’‘B’

‘S’

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Tree RecursionMethod countNodes (pseudocode):

if tree is nullreturn 0

else return countNodes(Left(tree)) +

countNodes(Right(tree)) + 1

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The Search Operation

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The Search Operation

public boolean search(T item) { return recSearch(root, item);} //helper: recursive method called by searchpublic boolean recSearch(BinaryTreeNode<T> tree, T item) { if(tree == null) return false; else { Comparable<T> temp = (Comparable<T>) tree.info; if (temp.compareTo(item) == 0) return true; else if (temp.compareTo(item) > 0) return recSearch(tree.lLink, item); else return recSearch(tree.rLink, item); }}

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The Insert Operation

Important: A new node is always inserted into its appropriate position in the binary search tree as a leaf.

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Insertions into a Binary Search Tree

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The Insert Operationpublic void insert(T item) { root = recInsert(root, item);} //helper: recursive method called by insertpublic BinaryTreeNode<T> recInsert(BinaryTreeNode<T> tree, T item) { if(tree == null) { //create new node tree = new BinaryTreeNode<T>(item); } else { Comparable<T> temp = (Comparable<T>) tree.info; if (temp.compareTo(item) == 0) { System.err.print("Already in - no duplicates."); return null; } else if (temp.compareTo(item) > 0) tree.lLink = recInsert(tree.lLink, item); else tree.rLink = recInsert(tree.rLink, item); } return tree;}

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The Delete Operation

The delete operation has 3 cases Node to be deleted is a leaf Node to be deleted has 1 child (left OR

right) Node to be deleted has 2 children

(nonempty left and right subtrees)

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Deleting a Leaf Node

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Deleting a Node with One Child

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Deleting a Node with Two Children (Q)

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The Delete Operation

Base Case: If item's key matches Info(tree), delete node

General Case: If item < Info(tree), delete(Left(tree), item);

else

delete(Right(tree), item).

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Code for Recursive Deletepublic void delete(T item) { root = recDelete(root, item);} //helper: recursive method called by deletepublic BinaryTreeNode<T> recDelete(BinaryTreeNode<T> tree, T item) { if(tree == null){ //empty tree System.err.println("Cannot delete from an empty tree."); return null; } else { Comparable<T> temp = (Comparable<T>) tree.info; if (temp.compareTo(item) > 0) tree.lLink = recDelete(tree.lLink, item); else if(temp.compareTo(item) < 0) tree.rLink = recDelete(tree.rLink, item); else if(tree.lLink != null && tree.rLink != null) {// 2 children tree.info = findMin(tree.rLink).info; //tree.info = findMax(tree.lLink).info; tree.rLink = removeMin(tree.rLink); } else if(root.lLink != null) //1 left child tree = tree.lLink; else if(root.rLink != null) //1 right child tree = tree.rLink; return tree; } }

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Code for Recursive Delete

//helper: method called by recDelete

protected BinaryTreeNode<T> findMin(BinaryTreeNode<T> tree) {

if(tree != null)

while(tree.lLink != null)

tree = tree.lLink;

return tree;

}

//helper: method called by recDelete

protected BinaryTreeNode<T> removeMin(BinaryTreeNode<T> tree) {

if(tree == null){ //empty tree

System.err.println("Cannot delete from an empty tree.");

return null;

}

else if(tree.lLink != null) {

tree.lLink = removeMin(tree.lLink);

return tree;

}

else

return tree.rLink;

}