Trees Main and Savitch Chapter 10

Binary Trees A binary tree has nodes, similar to nodes in a linked list structure. Data of one sort or another may be stored at each node. But it is the connections between the nodes which characterize a binary tree.

A Binary Tree of States In this example, the data contained at each node is one of the 50 states.

A Binary Tree of States Each tree has a special node called its root, usually drawn at the top.

A Binary Tree of States Each tree has a special node called its root, usually drawn at the top. The example tree has Washington as its root. The example tree has Washington as its root.

A Binary Tree of States Each node is permitted to have two links to other nodes, called the left child and the right child.

A Binary Tree of States Children are usually drawn below a node. The right child of Washington is Colorado. The right child of Washington is Colorado. The left child of Washington is Arkansas. The left child of Washington is Arkansas.

A Binary Tree of States Some nodes have only one child. Arkansas has a left child, but no right child. Arkansas has a left child, but no right child.

A Quiz Some nodes have only one child. Which node has only a right child? Which node has only a right child?

A Quiz Some nodes have only one child. Florida has only a right child. Florida has only a right child.

A Binary Tree of States A node with no children is called a leaf.

A Binary Tree of States Each node is called the parent of its children. Washington is the parent of Arkansas and Colorado. Washington is the parent of Arkansas and Colorado.

A Binary Tree of States Two rules about parents: ¶The root has no parent. ·Every other node has exactly one parent.

A Binary Tree of States Two nodes with the same parent are called siblings. Arkansas and Colorado are siblings. Arkansas and Colorado are siblings.

Complete Binary Trees A complete binary tree is a special kind of binary tree which will be useful to us.

Complete Binary Trees A complete binary tree is a special kind of binary tree which will be useful to us. When a complete binary tree is built, its first node must be the root. When a complete binary tree is built, its first node must be the root.

Complete Binary Trees The second node of a complete binary tree is always the left child of the root...

Complete Binary Trees The second node of a complete binary tree is always the left child of the root and the third node is always the right child of the root.

Complete Binary Trees The next nodes must always fill the next level from left to right.

Complete Binary Trees The next nodes must always fill the next level from left to right.

Complete Binary Trees The next nodes must always fill the next level from left to right.

Complete Binary Trees The next nodes must always fill the next level from left to right.

Complete Binary Trees The next nodes must always fill the next level from left to right.

Complete Binary Trees The next nodes must always fill the next level from left to right.

Is This Complete?

Yes! It is called the empty tree, and it has no nodes, not even a root.

Implementing a Complete Binary Tree We will store the data from the nodes in a partially-filled array. An array of data We don't care what's in this part of the array. An integer to keep track of how many nodes are in the tree 3

Implementing a Complete Binary Tree We will store the date from the nodes in a partially-filled array. An array of data We don't care what's in this part of the array. An integer to keep track of how many nodes are in the tree 3 Read Section 10.2 to see details of how the entries are stored. Read Section 10.2 to see details of how the entries are stored.

Depth of Binary Tree This example, depth = 3

Depth of Binary Tree This example, depth = 2

Depth of Binary Tree This example, depth = 0

Depth of Complete Binary Tree Given a complete binary tree of N nodes, what is the depth? D = 0 N = 1 D = 1 N = 3 D = 2 N = 7 D = 1 N = 2 D = 2 N = 4

Depth of Complete Binary Tree Given a complete binary tree of N nodes, what is the depth? D = 0 N = 1 D = 1 N = 3 D = 2 N = 7 D = 1 N = 2 D = 2 N = 4 D = floor(log N) = O(log N)

Depth of Binary Tree Given a binary tree of N nodes, what is the maximum possible depth? D = 0 N = 1 D = 2 N = 3 D = 4 N = 5 D = O(N)

Binary trees contain nodes. Each node may have a left child and a right child. If you start from any node and move upward, you will eventually reach the root. Every node except the root has one parent. The root has no parent. Complete binary trees require the nodes to fill in each level from left-to-right before starting the next level. Summary

One of the tree applications in Chapter 10 is binary search trees. In Chapter 10, binary search trees are used to implement bags and sets. This presentation illustrates how another data type called a dictionary is implemented with binary search trees. Binary Search Trees

The Dictionary Data Type A dictionary is a collection of items, similar to a bag. But unlike a bag, each item has a string attached to it, called the item's key.

The Dictionary Data Type A dictionary is a collection of items, similar to a bag. But unlike a bag, each item has a string attached to it, called the item's key. Example: The items I am storing are records containing data about a state.

The Dictionary Data Type A dictionary is a collection of items, similar to a bag. But unlike a bag, each item has a string attached to it, called the item's key. Example: The key for each record is the name of the state. Washington

The Dictionary Data Type The insertion procedure for a dictionary has two parameters. void Dictionary::insert(The key for the new item, The new item); Washington

The Dictionary Data Type When you want to retrieve an item, you specify the key... Item Dictionary::retrieve("Washington");

The Dictionary Data Type p When you want to retrieve an item, you specify the key and the retrieval procedure returns the item.

The Dictionary Data Type We'll look at how a binary tree can be used as the internal storage mechanism for the dictionary.

Arizona Arkansas Colorado A Binary Search Tree of States The data in the dictionary will be stored in a binary tree, with each node containing an item and a key. Washington Oklahoma Florida Mass. New Hampshire West Virginia

Colorado Arizona Arkansas A Binary Search Tree of States Washington Oklahoma Colorado Florida Mass. New Hampshire West Virginia Storage rules: 1.Every key to the left of a node is alphabetically before the key of the node.

Arizona Colorado Arkansas A Binary Search Tree of States Storage rules: 1.Every key to the left of a node is alphabetically before the key of the node. Washington Oklahoma Florida Mass. New Hampshire West Virginia Example: ' Massachusetts' and ' New Hampshire' are alphabetically before 'Oklahoma'

Arizona Arkansas A Binary Search Tree of States Storage rules: 1.Every key to the left of a node is alphabetically before the key of the node. 2.Every key to the right of a node is alphabetically after the key of the node. Washington Oklahoma Colorado Florida West Virginia Mass. New Hampshire

Arizona Arkansas A Binary Search Tree of States Storage rules: 1.Every key to the left of a node is alphabetically before the key of the node. 2.Every key to the right of a node is alphabetically after the key of the node. Washington Oklahoma Colorado Florida West Virginia Mass. New Hampshire

Arizona Arkansas Retrieving Data Start at the root. 1. If the current node has the key, then stop and retrieve the data. 2. If the current node's key is too large, move left and repeat If the current node's key is too small, move right and repeat 1-3. Washington Oklahoma Colorado Florida West Virginia Mass. New Hampshire

Arizona Arkansas Retrieve ' New Hampshire' Washington Oklahoma Colorado Florida West Virginia Mass. New Hampshire Start at the root. 1. If the current node has the key, then stop and retrieve the data. 2. If the current node's key is too large, move left and repeat If the current node's key is too small, move right and repeat 1-3.

Arizona Arkansas Retrieve 'New Hampshire' Washington Oklahoma Colorado Florida West Virginia Mass. New Hampshire Start at the root. 1. If the current node has the key, then stop and retrieve the data. 2. If the current node's key is too large, move left and repeat If the current node's key is too small, move right and repeat 1-3.

Arizona Arkansas Retrieve 'New Hampshire' Washington Oklahoma Colorado Florida West Virginia Mass. New Hampshire Start at the root. 1. If the current node has the key, then stop and retrieve the data. 2. If the current node's key is too large, move left and repeat If the current node's key is too small, move right and repeat 1-3.

Arizona Arkansas Retrieve 'New Hampshire' Washington Oklahoma Colorado Florida West Virginia Mass. New Hampshire Start at the root. 1. If the current node has the key, then stop and retrieve the data. 2. If the current node's key is too large, move left and repeat If the current node's key is too small, move right and repeat 1-3.

Retrieval: Complexity Given a complete tree of N items, the depth D = O(log N). What is the maximum number of nodes tested to retrieve an item from the binary search tree if we use a complete tree? What is the maximum number of nodes tested (worst case) to retrieve an item from a binary search tree that is not complete or balanced?

Arizona Arkansas Adding a New Item with a Given Key 1. Pretend that you are trying to find the key, but stop when there is no node to move to. 2. Add the new node at the spot where you would have moved to if there had been a node. Washington Oklahoma Colorado Florida West Virginia Mass. New Hampshire

Arizona Arkansas Adding 1.Pretend that you are trying to find the key, but stop when there is no node to move to. 2.Add the new node at the spot where you would have moved to if there had been a node. Washington Oklahoma Colorado Florida West Virginia Mass. New Hampshire Iowa

Arizona Arkansas Adding 1.Pretend that you are trying to find the key, but stop when there is no node to move to. 2.Add the new node at the spot where you would have moved to if there had been a node. Washington Oklahoma Colorado Florida West Virginia Mass. New Hampshire Iowa

Arizona Arkansas Adding 1.Pretend that you are trying to find the key, but stop when there is no node to move to. 2.Add the new node at the spot where you would have moved to if there had been a node. Washington Oklahoma Colorado Florida West Virginia Mass. New Hampshire Iowa

Arizona Arkansas Adding 1.Pretend that you are trying to find the key, but stop when there is no node to move to. 2.Add the new node at the spot where you would have moved to if there had been a node. Washington Oklahoma Colorado Florida West Virginia Mass. New Hampshire Iowa

Arizona Arkansas Adding 1.Pretend that you are trying to find the key, but stop when there is no node to move to. 2.Add the new node at the spot where you would have moved to if there had been a node. Washington Oklahoma Colorado Florida West Virginia Mass. New Hampshire Iowa

Arizona Arkansas Adding 1.Pretend that you are trying to find the key, but stop when there is no node to move to. 2.Add the new node at the spot where you would have moved to if there had been a node. Washington Oklahoma Colorado Florida West Virginia Mass. New Hampshire Iowa

Arizona Arkansas Adding Washington Oklahoma Colorado Florida West Virginia Mass. New Hampshire Iowa Where would you add this state? Where would you add this state? Kazakhstan

Arizona Arkansas Adding Washington Oklahoma Colorado Florida West Virginia Mass. New Hampshire Iowa Kazakhstan is the new right child of Iowa? Kazakhstan is the new right child of Iowa? Kazakhstan

Adding: Complexity Given a complete tree of N items, the depth D = O(log N). What is the maximum number of nodes tested to add an item to the binary search tree if we use a complete tree? What is the maximum number of nodes tested (worst case) to add an item from a binary search tree that is not complete or balanced?

Arizona Arkansas Removing an Item with a Given Key 1.Find the item. 2.If necessary, swap the item with one that is easier to remove. 3.Remove the item. Washington Oklahoma Colorado Florida West Virginia Mass. New Hampshire Iowa Kazakhstan

Arizona Arkansas Removing 'Florida' 1.Find the item. Washington Oklahoma Colorado Florida West Virginia Mass. New Hampshire Iowa Kazakhstan

Arizona Arkansas Removing 'Florida' Washington Oklahoma Colorado Florida West Virginia Mass. New Hampshire Iowa Kazakhstan Florida cannot be removed at the moment... Florida cannot be removed at the moment...

Arizona Arkansas Removing 'Florida' Washington Oklahoma Colorado West Virginia Mass. New Hampshire Iowa Kazakhstan... because removing Florida would break the tree into two pieces.... because removing Florida would break the tree into two pieces.

Arizona Arkansas Removing 'Florida' Washington Oklahoma Colorado Florida West Virginia Mass. New Hampshire Iowa Kazakhstan The problem of breaking the tree happens because Florida has 2 children. The problem of breaking the tree happens because Florida has 2 children. 1.If necessary, do some rearranging.

Arizona Arkansas Removing 'Florida' If necessary, do some rearranging. Washington Oklahoma Colorado Florida West Virginia Mass. New Hampshire Iowa Kazakhstan For the rearranging, take the smallest item in the right subtree... For the rearranging, take the smallest item in the right subtree...

Arizona Arkansas Removing 'Florida' Washington Oklahoma Colorado West Virginia Mass. New Hampshire Iowa Kazakhstan Iowa...copy that smallest item onto the item that we're removing......copy that smallest item onto the item that we're removing... If necessary, do some rearranging.

Arizona Arkansas Removing 'Florida' Washington Oklahoma Colorado West Virginia Mass. New Hampshire Iowa Kazakhstan... and then remove the extra copy of the item we copied and then remove the extra copy of the item we copied... If necessary, do some rearranging.

Arizona Arkansas Removing 'Florida' Washington Oklahoma Colorado West Virginia Mass. New Hampshire Iowa Kazakhstan... and reconnect the tree If necessary, do some rearranging.

Arizona Arkansas Removing 'Florida' Washington Oklahoma Colorado Florida West Virginia Mass. New Hampshire Kazakhstan Why did we choose the smallest item in the right subtree? Why did we choose the smallest item in the right subtree?

Arizona Arkansas Removing 'Florida' Washington Oklahoma Colorado West Virginia Mass. New Hampshire Iowa Kazakhstan Because every key must be smaller than the keys in its right subtree

Removing an Item with a Given Key 1.Find the item. 2.If the item has a right child, rearrange the tree: 1.Find smallest item in the right subtree 2.Copy that smallest item onto the one that you want to remove 3.Remove the extra copy of the smallest item (making sure that you keep the tree connected) 3. else just remove the item.

Binary search trees are a good implementation of data types such as sets, bags, and dictionaries. Searching for an item is generally quick since you move from the root to the item, without looking at many other items. Adding and deleting items is also quick. But as you'll see later, it is possible for the quickness to fail in some cases -- can you see why? Summary