1.1 Data Structure and Algorithm Lecture 12 Binary Search Trees Topics Reference: Introduction to Algorithm by Cormen Chapter 13: Binary Search Trees

1.2 Data Structure and Algorithm subtree Trees Node Edge parent root leaf interior node child path Degree? Depth/Level? Height?

1.3 Data Structure and Algorithm Binary Tree Representation Node  data  left child  right child  parent (optional) Parent Data Left Right Tree root

1.4 Data Structure and Algorithm Full Binary Tree  Replace each missing child in the binary tree with a node having no children (black nodes) which are called leaf nodes.  Each node is either a leaf or has degree exactly 2  All leaves are at level h (height)  All interior nodes are full Full Binary Tree Binary Tree

1.5 Data Structure and Algorithm Complete Binary Tree  All leaves have the same depth  All interior nodes have degree 2 depth 0 depth 1 depth 2 depth 3 height=3

1.6 Data Structure and Algorithm Complete Binary Tree The number of internal nodes of a complete binary tree of height h is: ……2 h-1 = 2 h -1/2-1 = 2 h -1 What is the number of total nodes in a complete binary tree of height h? What is the number of leaf nodes in a complete binary tree of height h?

1.7 Data Structure and Algorithm Applications - Expression Trees * (5*3)+(8-4) To represent infix expressions

1.8 Data Structure and Algorithm Applications - Parse Trees Used in compilers to check syntax statement ifcondthen statementelsestatement ifcondthen statement

1.9 Data Structure and Algorithm Binary Search Trees Binary Search Trees (BSTs) are an important data structure for dynamic sets In addition to data, elements have:  key: an identifying field inducing a total ordering  left: pointer to a left child (may be NULL)  right: pointer to a right child (may be NULL)  p: pointer to a parent node (NULL for root)

1.10 Data Structure and Algorithm Binary Search Trees BST property: key[leftSubtree(x)]  key[x]  key[rightSubtree(x)] Example: F BH KDA

1.11 Data Structure and Algorithm Traversals for a Binary Tree Pre order  visit the node  go left  go right In order  go left  visit the node  go right Post order  go left  go right  visit the node

1.12 Data Structure and Algorithm Traversal Examples A B D G C E H F I Pre order A B D G H C E F I In order G D H B A E C F I Post order G H D B E I F C A

1.13 Data Structure and Algorithm Traversal Implementation recursive implementation of preorder  pre-order ( node )  visit node  pre-order ( node.left )  pre-order ( node.right ) What changes need to be made for in-order, post-order?

1.14 Data Structure and Algorithm Inorder Tree Walk n in-order(x) in-order (x.left); print(x); in-order(x.right); Prints elements in sorted (increasing) order

1.15 Data Structure and Algorithm Postorder Tree Walk n post-order(x) print(x) post-order (x.left); post-order(x.right);

1.16 Data Structure and Algorithm Evaluating an expression tree Walk the tree in postorder When visiting a node, use the results of its children to evaluate it *

1.17 Data Structure and Algorithm BST Operations Search  Key  Minimum  Maximum  Successor  Predecessor Insert node Delete node

1.18 Data Structure and Algorithm BST Search: Key Search for D and C: F BH KDA

1.19 Data Structure and Algorithm BST Search: Key Given a key and a pointer to a node, returns an element with that key or NULL: TreeSearch(x, k) if (x = NULL or k = key[x]) return x; if (k < key[x]) return TreeSearch(left[x], k); else return TreeSearch(right[x], k); Cost: O(lg 2 n) Where height, h=lg 2 n

1.20 Data Structure and Algorithm BST Search: Minimum TreeMinimum(x)  While left[x] <> NIL do  x = left[x]  return x

1.21 Data Structure and Algorithm BST Search: Maximum TreeMaximum(x)  While right[x] <> NIL do  x = right[x]  return x

1.22 Data Structure and Algorithm BST Search: Successor Inorder traversal: Case1:x has a right subtree successor is minimum node in right subtree Case2: x has no right subtree successor is first ancestor of x whose left child is also ancestor of x

1.23 Data Structure and Algorithm BST Search: Successor if right[x] <> NIL then  return TreeMinimum (right[x]) y = p[x] while y <> NIL and x = right[y] do  x = y  y = p[y] return y Predecessor: similar algorithm

1.24 Data Structure and Algorithm BST: Insert Node Example: Insert C F BH KDA C

1.25 Data Structure and Algorithm BST: Insert Node Adds an element x to the tree so that the binary search tree property continues to hold TreeInsert (T, z)  y = NIL  x = root[T]  While x <> NILL do  y = x  if key[z] < key[x] then x = left[x]  else x = right[x]  p[z] = y  if y == NIL then root[T] = z  else if key[z] <key[y] then left[y] = z  else right[y] = z Cost: O(lg 2 n) Where height, h=lg 2 n

1.26 Data Structure and Algorithm BST: Delete Node Deletion is a bit tricky 3 cases:  x has no children:  Remove x  x has one child:  Splice out x  x has two children:  Swap x with successor  Perform case 1 or 2 to delete it F BH KDA C Example: delete K or H or B