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Binary Trees

Binary Trees. A binary tree is made up of a finite set of nodes that is either empty or consists of a node called the root together with two binary trees called the left and right subtrees which are disjoint from each other and from the root

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Binary Trees

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  1. Binary Trees A binary tree is made up of a finite set of nodes that is either empty or consists of a node called the root together with two binary trees called the left and right subtrees which are disjoint from each other and from the root Notation: Node, Children, Edge, Parent, Ancestor, Descendant, Path, Depth, Height, Level, LeafNode, Internal Node, Subtree. A B C D E F G H I

  2. Full and Complete Binary Trees A Full binary tree has each node being either a leaf or internal node with exactly two non-empty children. A Complete binary tree: If the height of the tree is d, then all levels except possibly level d are completely full. The bottom level has all nodes to the left side.

  3. Full Binary Tree Theorem Theorem: The number of leaves in a non-empty full binary tree is one more than the number of internal nodes. Proof (by Mathematical Induction) • Base Case: A full binary tree with 1 internal node must have two leaf nodes. • Induction Hypothesis: Assume any full binary tree T containing n-1 internal nodes has n leaves. • Induction Step: Pick an arbitrary leaf node j of T. Make j an internal node by giving it two children. The number of internal nodes has now gone up by 1 to reach n. The number of leaves has also gone up by 1. Corollary: The number of NULL pointers in a non-empty binary tee is one more than the number of nodes in the tree.

  4. Traversals Any process for visiting the nodes in some order is called a traversal. Any traversal that lists every node in the tree exactly once is called an enumeration of the tree’s nodes. Preorder traversal: Visit each node before visiting its children. Postorder traversal: Visit each node after visiting its children. Inorder traversal: Visit the left subtree, then the node, then the right subtree. void preorder(BinNode * root) { if (root==NULL) return; visit(root); preorder(root->leftchild()); preorder(root->rightchild()); }

  5. Expression Trees • Example of (a-b)/((x*y+3)-(6*z)) / - - a b + * 6 z * 3 x y

  6. Binary Search Trees • Left means less – right means greater. • Find • If item<cur->dat then cur=cur->left • Else if item>cur->dat then cur=cur->right • Else found • Repeat while not found and cur not NULL • No need for recursion.

  7. Find min and max • The min will be all the way to the left • While cur->left != NULL, cur=cur->left • The max will be all the way to the right • While cur->right !=NULL, cur=cur->right • Insert • Like a find, but stop when you would go to a null and insert there.

  8. Remove • If node to be deleted is a leaf (no children), can remove it and adjust the parent node (must keep track of previous) • If the node to be deleted has one child, remove it and have the parent point to that child. • If the node to be deleted has 2 children • Replace the data value with the smallest value in the right subtree • Delete the smallest value in the right subtree (it will have no left child, so a trivial delete.

  9. 0 1 2 3 4 5 6 7 8 9 10 11 Array Implementation • For a complete binary tree • Parent(x)= • Leftchild(x)= • Rightchild(x)= • Leftsibling(x)= • Rightsibling(x)= (x-1)/2 2x+1 2x+2 x-1 x+1

  10. Huffman Coding Trees • Each character has exactly 8 bits • Goal: to have a message/file take less space • Allow some characters to have shorter bit patterns, but some characters can have longer. • Will not have any benefit if each character appears with equal probability. • English does not have equal distribution of character occurance. • If we let the single bit 1 represent the letter E, then no other character can start with a 1. So some will have to be longer.

  11. The Tree • Look at the left pointer as the way to go for a 0 and the right pointer is the way to go for a 1. • Take input string of 1’s and 0’s and follow them until hit null pointer, then the character in that node is the character being held. • Example: • 0 = E • 10 = T • 110 = P • 111 = F • 0110111010=EPFET E T F P

  12. Weighted Tree • Each time we have to go to another level, takes time. Want to go down as few times as needed. • Have the most frequently used items at the top, least frequently items at the bottom. • If we have weights or frequencies of nodes, then we want a tree with minimal external path weight.

  13. Huffman Example • Assume the following characters with their relative frequencies. Z K F C U D L E 2 7 24 32 37 42 42 120 • Arrange from smallest to largest. • Combine 2 smallest with a parent node with the sum of the 2 frequencies. • Replace the 2 values with the sum. Combine the nodes with the 2 smallest values until only one node left.

  14. Huffman Tree Construction Z K F C U D L E 2 7 24 32 37 42 42 120 306 186 120 E 79 107 65 37 U 42 D 42 L 32 C 33 9 24 F 2 Z 7 K

  15. Results Let Freq Code Bits Mess. Len. Old Bits Old Mess. Len. C 32 1110 4 128 3 96 D 42 101 3 126 3 126 E 120 0 1 120 3 360 F 24 11111 5 120 3 72 K 7 111101 6 42 3 21 L 42 110 3 126 3 126 U 37 100 3 111 3 111 Z 2 111100 6 12 3 6 TOTAL: 785 918

  16. Heap • Is a complete binary tree with the heap property • min-heap : All values are less than the child values. • Max-heap: all values are greater than the child values. • The values in a heap are partially ordered. There is a relationship between a node’s value and the value of it’s children nodes. • Representation: Usually the array based complete binary tree representation.

  17. Building the Heap • Several ways to build the heap. As we add each node, or create “a” tree as we get all the data and then “heapify” it. • More efficient if we wait until all data is in. 1 1 2 3 5 7 4 5 6 7 4 2 6 3 7 7 5 1 5 6 4 2 6 3 4 2 1 3

  18. Heap ADT class heap{ private: ELEM* Heap; int size; int n; void siftdown(int); public: heap(ELEM*, int, int); int heapsize() const; bool isLeaf(int) const; int leftchild(int) const; int rightchild(int) const; int parent(int) const; void insert(const ELEM); ELEM removemax(); ELEM remove(int); void buildheap();};

  19. Siftdown For fast heap construction • Work from high end of array to low end. • Call siftdown for each item. • Don’t need to call siftdown on leaf nodes. void heap::buildheap() {for (int i=n/2-1; i>=0; i--) siftdown(i);} void heap::siftdown(int pos){ assert((pos>=0) && (pos<n)); while (!isleaf(pos)) { int j=leftchild(pos); if ((j<(n-1) && key(Heap[j])<key(Heap[j+1]))) j++; // j now has position of child with greater value if (key(Heap[pos])>=key(Heap[j])) return; swap(Heap[pos], Heap[j]); pos=j; } }

  20. Priority Queues • A priority queue stores objects and on request, releases the object with the greatest value. • Example : scheduling jobs in a multi-tasking operating system. • The priority of a job may change, requiring some reordering of the jobs. • Implementation: use a heap to store the priority queue. • To support priority reordering, delete and re-insert. Need to know index for the object. ELEM heap::remove(int pos) { assert((pos>0) && (pos<n)); swap (Heap[pos], Heap[--n]); if (n!=0) siftdown(pos); return Heap[n]; }

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