1 A Two-Level Binary Expression ‘-’ ‘8’ ‘5’ treePtr INORDER TRAVERSAL : has value 3 PREORDER TRAVERSAL: POSTORDER TRAVERSAL: 8 5 -

2 Levels Indicate Precedence When a binary expression tree is used to represent an expression, the levels of the nodes in the tree indicate their relative precedence of evaluation. Operations at higher levels of the tree are evaluated later than those below them. The operation at the root is always the last operation performed.

3 A Binary Expression Tree ‘*’ ‘+’ ‘4’ ‘3’ ‘2’ What value does it have? ( ) * 3 = 18 What infix, prefix, postfix expressions does it represent?

4 Inorder Traversal: (A + H) / (M - Y) ‘/’ ‘+’ ‘A’ ‘H’ ‘-’ ‘M’‘Y’ tree Print left subtree firstPrint right subtree last Print second

5 Preorder Traversal: / + A H - M Y ‘/’ ‘+’ ‘A’ ‘H’ ‘-’ ‘M’‘Y’ tree Print left subtree secondPrint right subtree last Print first

6 ‘/’ ‘+’ ‘A’ ‘H’ ‘-’ ‘M’‘Y’ tree Print left subtree firstPrint right subtree second Print last Postorder Traversal: A H + M Y - /

7 Evaluate: ‘*’ ‘-’ ‘8’ ‘5’ What infix, prefix, postfix expressions does it represent? ‘/’ ‘+’ ‘4’ ‘3’ ‘2’

8 Answer Infix: ( ( ) * ( ( ) / 3 ) ) Prefix: * / Postfix: / * has operators in order used ‘*’ ‘-’ ‘8’ ‘5’ ‘/’ ‘+’ ‘4’ ‘3’ ‘2’

9 InfoNode has 2 forms enum OpType { OPERATOR, OPERAND } ; struct InfoNode { OpType whichType; union// ANONYMOUS union { char operation ; int operand ; } };. whichType. operation OPERATOR ‘+’. whichType. operand OPERAND 7

10 TreeNodes struct TreeNode { InfoNode info ; // Data member TreeNode* left ; // Pointer to left child TreeNode* right ; // Pointer to right child };. left. info. right NULL whichType. operand OPERAND 7

11 class ExprTree ‘*’ ‘+’ ‘4’ ‘3’ ‘2’ ExprTree ~ExprTree Build Evaluate. private: root

int Eval ( TreeNode* ptr ) // Pre: ptr is a pointer to a binary expression tree. // Post: Function value = the value of the expression represented // by the binary tree pointed to by ptr. { switch ( ptr->info.whichType ) { case OPERAND : return ptr->info.operand ; case OPERATOR : switch ( tree->info.operation ) { case ‘+’ : return ( Eval ( ptr->left ) + Eval ( ptr->right ) ) ; case ‘-’ : return ( Eval ( ptr->left ) - Eval ( ptr->right ) ) ; case ‘*’ : return ( Eval ( ptr->left ) * Eval ( ptr->right ) ) ; case ‘/’ : return ( Eval ( ptr->left ) / Eval ( ptr->right ) ) ; } 12

13 A Full Binary Tree A full binary tree is a binary tree in which all the leaves are on the same level and every non leaf node has two children. SHAPE OF A FULL BINARY TREE

14 A Complete Binary Tree A complete binary tree is a binary tree that is either full or full through the next-to-last level, with the leaves on the last level as far to the left as possible. SHAPE OF A COMPLETE BINARY TREE

15 What is a Heap? A heap is a binary tree that satisfies these special SHAPE and ORDER properties: n Shape: a complete binary tree. n Order: the value stored in each node is greater than or equal to the value in each of its children.

16 Are These Both Heaps? C A T treePtr

tree Is This a Heap?

tree Where is the Largest Element in a Heap?

tree Number the Nodes Left to Right by Level

tree Use the Numbers as Array Indexes to Store the Tree [ 0 ] [ 1 ] [ 2 ] [ 3 ] [ 4 ] [ 5 ] [ 6 ] tree.nodes

// HEAP SPECIFICATION // Assumes ItemType is either a built-in simple data type // or a class with overloaded realtional operators. template struct HeapType { void ReheapDown ( int root, int bottom ) ; void ReheapUp ( int root, int bottom ) ; ItemType* elements ; // ARRAY to be allocated dynamically int numElements ; }; 21

22 Observations l Heap is a struct with member functions l Why not a class? n To allow access to its components n Heaps are usually used in conjunction with other classes which want access to the components but will hide them from the application

23 Reheap Down l For heaps with all but the top element in heap position l How: n Swap root with left or right child n Reheap (down) with that child l Used to remove the highest element: n Remove the root n Put last element in root position n Reheap down to restore the order

24 Reheap Up l For a node that violates a heap up: swap with parent until it is a heap l So, if we added a node to the “end” of the tree, reheap up would put it in the right spot l To build a heap: n start with empty heap n add element to the end and reheap up

25 Exercises l Make a heap out of elements D H A K P R l Remove the top element and reheap.

26 // IMPLEMENTATION OF RECURSIVE HEAP MEMBER FUNCTIONS template void HeapType ::ReheapDown ( int root, int bottom ) // Pre: root is the index of the node that may violate the heap // order property // Post: Heap order property is restored between root and bottom { int maxChild ; int rightChild ; int leftChild ; leftChild = root * ; rightChild = root * ; 26 ReheapDown

if ( leftChild <= bottom )// ReheapDown continued { if ( leftChild == bottom ) maxChild = leftChild ; else { if (elements [ leftChild ] <= elements [ rightChild ] ) maxChild = rightChild ; else maxChild = leftChild ; } if ( elements [ root ] < elements [ maxChild ] ) { Swap ( elements [ root ], elements [ maxChild ] ) ; ReheapDown ( maxChild, bottom ) ; } 27

// IMPLEMENTATIONcontinued template void HeapType ::ReheapUp ( int root, int bottom ) // Pre: bottom is the index of the node that may violate the heap // order property. The order property is satisfied from root to // next-to-last node. // Post: Heap order property is restored between root and bottom { int parent ; if ( bottom > root ) { parent = ( bottom - 1 ) / 2; if ( elements [ parent ] < elements [ bottom ] ) { Swap ( elements [ parent ], elements [ bottom ] ) ; ReheapUp ( root, parent ) ; } 28

29 Priority Queue A priority queue is an ADT with the property that only the highest-priority element can be accessed at any time. Example: homework assignments are ordered by priority. Like a sorted list except for the access restriction. Goal: be able to insert in order and to access the top element efficiently.

ADT Priority Queue Operations Transformers n MakeEmpty n Enqueue n Dequeue Observers n IsEmpty n IsFull change state observe state 30

// CLASS PQTYPE DEFINITION AND MEMBER FUNCTIONS // template class PQType { public: PQType( int ); ~PQType ( ); void MakeEmpty( ); bool IsEmpty( ) const; bool IsFull( ) const; void Enqueue( ItemType item ); void Dequeue( ItemType& item ); private: int numItems; HeapType items; int maxItems; }; 31

PQType ~PQType Enqueue Dequeue. class PQType [0] [1] [2] [3] [4] [5] [6] [7] [8] [9] ‘X’ ‘C’ ‘J’ Private Data: numItems 3 maxItems 10 items.elements.numElements

33 Graphs A data structure that consists of a set of nodes (vertices) and arcs (edges) that relate the nodes to each other. G = (V, E) E.g. a map with V = cities and E = roads Undirected versus directed: some of the streets may be one-way. Weighted: each edge has a value (miles).

Applications Traversal: Find a route (path) from Boston to Chicago. Find the shortest path. Approaches: Depth-first search Breadth-first search 34

35 Implementation via Adjacency Matrices l For a graph with N vertices: numVertices = the number of nodes vertices[N] = name of vertex N edges[N] [M] = weight from vertex N to M l See p l How do you add a vertex? An edge? l How do you find a path?

36 Implementation via Adjacency Lists l An array of vertices. l Each vertex has a linked list of the nodes extending directly from it. l See p l When would you use this? When not?