Paul Tymann, Andrew Watkins, Tree Traversal Section 9.3 Longin Jan Latecki Temple University Based on slides by Paul Tymann, Andrew Watkins, and J. van Helden

Tree Anatomy The children of a node are, themselves, trees, called subtrees. Root Level 0 R Level 1 S T Internal Node Level 2 X U V W Leaf Level 3 Y Z Child of X Subtree Parent of Z and Y

Tree Traversals One of the most common operations performed on trees, are a tree traversals A traversal starts at the root of the tree and visits every node in the tree exactly once visit means to process the data in the node Traversals are either depth-first or breadth-first

Breadth First Traversals All the nodes in one level are visited Followed by the nodes the at next level Beginning at the root For the sample tree 7, 6, 10, 4, 8, 13, 3, 5 7 6 10 4 8 13 3 5

Queue and stack A queue is a sequence of elements such that each new element is added (enqueued) to one end, called the back of the queue, and an element is removed (dequeued) from the other end, called the front A stack is a sequence of elements such that each new element is added (or pushed) onto one end, called the top, and an element is removed (popped) from the same end

Breadth first tree traversal with a queue Enqueue root While queue is not empty Dequeue a vertex and write it to the output list Enqueue its children left-to-right Step Output Queue 0 a 1 a e,d 2 e d,i,b 3 d i,b,k,l 4 i b,k,l 5 b k,l,f 6 k l,f 7 l f 8 f g 9 g j,h 10 j h,m 11 h m 12 m c 13 c a b i f e d l g j h c m k 1 5 4 8 2 3 7 9 10 11 13 12 6

Depth-First Traversals There are 6 different depth-first traversals VLR (pre-order traversal) VRL LVR (in-order traversal) RVL RLV LRV (post-order traversal)

Pre-order Traversal: VLR Visit the node Do a pre-order traversal of the left subtree Finish with a pre-order traversal of the right subtree For the sample tree 7, 6, 4, 3, 5, 10, 8, 13 7 6 10 4 8 13 3 5

Pre-order tree traversal with a stack Push root onto the stack While stack is not empty Pop a vertex off stack, and write it to the output list Push its children right-to-left onto stack Step Output Stack 0 a 1 a d,e 2 e d,b,i 3 i d,b 4 b d,f 5 f d,g 6 g d,h,j 7 j d,h,m 8 m d,h,c 9 c d,h 10 h d 11 d l,k 12 k l 13 l a b i f e d l g j h c m k 1 4 3 5 2 11 13 6 7 10 9 8 12

Preorder Traversal Step 1: Visit r Step 2: Visit T1 in preorder Tn Step 2: Visit T1 in preorder Step 3: Visit T2 in preorder Step n+1: Visit Tn in preorder

Example A R E Y P M H J Q T M A J Y R H P Q T E

Ordering of the preorder traversal is the same a the Universal Address System with lexicographic ordering. A R E Y P M H J Q T 1 2 3 2.2 2.1 1.1 2.2.1 2.2.2 2.2.3 M A J Y R H P Q T E

In-order Traversal: LVR Do an in-order traversal of the left subtree Visit the node Finish with an in-order traversal of the right subtree For the sample tree 3, 4, 5, 6, 7, 8, 10, 13 7 6 10 4 8 13 3 5

Inorder Traversal Step 1: Visit T1 in inorder Step 2: Visit r Tn Step 2: Visit r Step 3: Visit T2 in inorder Step n+1: Visit Tn in inorder

Example A R E Y P M H J Q T J A M R Y P H Q T E

inorder (t) if t != NIL: { inorder (left[t]); write (label[t]); inorder (right[t]); } Inorder Traversal on a binary search tree.

Post-order Traversal: LRV Do a post-order traversal of the left subtree Followed by a post-order traversal of the right subtree Visit the node For the sample tree 3, 5, 4, 6, 8, 13, 10, 7 7 6 10 4 8 13 3 5

Postorder Traversal Step 1: Visit T1 in postorder Tn Step 2: Visit T2 in postorder Step n: Visit Tn in postorder Step n+1: Visit r

Example A R E Y P M H J Q T J A R P Q T H Y E M

Representing Arithmetic Expressions Complicated arithmetic expressions can be represented by an ordered rooted tree Internal vertices represent operators Leaves represent operands Build the tree bottom-up Construct smaller subtrees Incorporate the smaller subtrees as part of larger subtrees

Example (x+y)2 + (x-3)/(y+2) + 2  / + x y – x 3 + y 2

Infix Notation Traverse in inorder (LVR) adding parentheses for each operation +  – / 2 x y 3 ( ) ( ) ( ) x + y  2 + ( ) ( ) x – 3 / ( ) y + 2

Prefix Notation (Polish Notation) Traverse in preorder (VLR) +  – / 2 x y 3 +  + x y 2 / – x 3 + y 2

Evaluating Prefix Notation In an prefix expression, a binary operator precedes its two operands The expression is evaluated right-left Look for the first operator from the right Evaluate the operator with the two operands immediately to its right

Example + / + 2 2 2 / – 3 2 + 1 0 + / + 2 2 2 / – 3 2 1 + / + 2 2 2 / 1 1 + / + 2 2 2 1 + / 4 2 1 + 2 1 3

Postfix Notation (Reverse Polish) Traverse in postorder (LRV) +  – / 2 x y 3 x y + 2  x 3 – y 2 + / +

Evaluating Postfix Notation In an postfix expression, a binary operator follows its two operands The expression is evaluated left-right Look for the first operator from the left Evaluate the operator with the two operands immediately to its left

Example 2 2 + 2 / 3 2 – 1 0 + / + 4 2 / 3 2 – 1 0 + / + 2 3 2 – 1 0 + / + 2 1 1 0 + / + 2 1 1 / + 2 1 + 3