Data Structures – Binary Tree

What is a tree?

Where do you see trees? Ummm...outside Family trees File systems

Tree Terminology node – any element of the tree root – the topmost node of the tree parent – a node that has one or more nodes connected below it (children) child – a node that has a connected node above it (parent) leaf – any child node at the bottom of the tree subtree – any node and all its descendants

Name that part! root / parent subtree parent / child’ child / leaf

So what’s a binary tree? A parent can have at most TWO children (left child, right child) The left child’s data and all the data in the left subtree is “less than” the parent’s data The right child’s data all the data in the right subtree is “greater than” the parent’s data NOTE: The “less than” and “greater than” requirements of the subtrees is commonly known outside of the IB world as a Binary Search Tree

Example – Valid Binary Tree 6 2 10 4 1

Example – INVALID Binary Tree 3 2 10 4 1

Adding to a Binary Tree Start at the root Compare against the current node Go left if less than current node OR insert if there is none (creating a new leaf) Go right if greater than current node OR insert if there is none (creating a new leaf) Repeat step 2

Creating a binary tree Insert the following numbers in the given order: 6, 4, 2, 7, 8, 14, 9

Your Turn Create a binary tree by inserting the following numbers in the given order : 14, 75, 2, 5, 60, 25, 1, 9

Your Turn Again Create a binary tree by inserting the following numbers in the given order : 2, 4, 6, 8, 10 Notice anything weird? It’s unbalanced! 

Another Binary Tree h d m a k x

Practice w/ Strings Create a binary tree by inserting the following strings: Wanda, Alpos, Vazbyte, Fecso, Downs, Mata, Montante

Searching in a Binary Tree Start at the root Compare against the current node Found the node if they are equal Go left if less than current node OR if there is no left, then does not exist Go right if greater than current node OR if there is no left, then does not exist Repeat step 2

Search(4) – What path do you take? 6 2 10 4 1 13

Search(9) – What path do you take? 6 2 10 4 1 13

Removing from a Binary Tree Search for matching node If the node is a leaf (0 children), unlink its parent If the node has 1 child, then link the node’s parent to the node’s child If the node has 2 children, then go to its right subtree and find the left-most child (smallest value of the right subtree). Take the value and put it in the original node that was being removed. Repeat the remove process on the node with 2 children

Remove(4) – 0 children case 6 2 10 4 1 13

Remove(10) – 1 child case 6 2 10 4 1 13

Remove(6) – 2 children case 10 4 1 13

When do we use binary trees? ...whenever we need to search for quickly Inherent binary searching capabilities Large trees can be searched quickly What is the best case scenario? What is the worst case scenario? Balanced tree? Unbalanced tree? What is the average case scenario? Compare a Linked List to a Binary Tree

Tree Traversal Add, remove, search only go down 1 path How do you “walk-through” all the nodes of a tree?

In-order tree traversal Left-subtree traversal if it exists and rerun Action on the current node (e.g. print) Right-subtree traversal if it exists and rerun Note: In-order traversal often used to visit nodes in their inherent order

In-order Traversal (1, 2, 4, 6, 8, 10, 13) 6 2 10 4 1 8 13

Pre-order tree traversal Action on the current node (e.g. copy) Left-subtree traversal if it exists and rerun Right-subtree traversal if it exists and rerun Note: Pre-order traversal is often used to duplicate a tree

Pre-order Traversal (6, 2, 1, 4, 10, 8, 13)

Post-order tree traversal Left-subtree traversal if it exists and rerun Right-subtree traversal if it exists and rerun Action on the current node (e.g. print) Note: Post-order traversal is often used to completely delete or free up all nodes by visiting children and lowest levels first. (not really necessary with garbage collection)

Post-order Traversal (1, 4, 2, 8, 13, 10, 6)

Other Resources http://www.csanimated.com/animation.php?t=Binary_search_tree