Tree Traversal • Process of visiting nodes in a tree systematically CSED233: Data Structures (2013F) . Some algorithms need to visit all nodes in a tree. Example: printing, counting nodes, etc. Lecture12: Trees II • Implementation . Can be done easily by recursion Computers”R”Us . Order of visits does matter. Bohyung Han Sales Manufacturing R&D CSE, POSTECH [email protected] US International Laptops Desktops Europe Asia Canada CSED233: Data Structures 2 by Prof. Bohyung Han, Fall 2013 Preorder Traversal Preorder Traversal • A preorder traversal of the subtree rooted at node n: . Visit node n (process the node's data). 1 . Recursively perform a preorder traversal of the left child. Recursively perform a preorder traversal of the right child. 2 7 • A preorder traversal starts at the root. public void preOrderTraversal(Node n) 3 6 8 12 { if (n == null) return; 4 5 9 System.out.print(n.value+" "); preOrderTraversal(n.left); preOrderTraversal(n.right); 10 11 } CSED233: Data Structures CSED233: Data Structures 3 by Prof. Bohyung Han, Fall 2013 4 by Prof. Bohyung Han, Fall 2013 Inorder Traversal Inorder Traversal • A preorder traversal of the subtree rooted at node n: . Recursively perform a preorder traversal of the left child. 6 . Visit node n (process the node's data). Recursively perform a preorder traversal of the right child. 4 11 • An iorder traversal starts at the root. public void inOrderTraversal(Node n) 2 5 7 12 { if (n == null) return; 1 3 9 inOrderTraversal(n.left); System.out.print(n.value+" "); inOrderTraversal(n.right); } 8 10 CSED233: Data Structures CSED233: Data Structures 5 by Prof. Bohyung Han, Fall 2013 6 by Prof. Bohyung Han, Fall 2013 Print Arithmetic Expressions Postorder Traversal • Specialization of an inorder traversal • A preorder traversal of the subtree rooted at node n: . print operand or operator when visiting node . Recursively perform a preorder traversal of the left child. print “(“ before traversing left subtree . Recursively perform a preorder traversal of the right child. print “)“ after traversing right subtree . Visit node n (process the node's data). 6 • A postorder traversal starts at the root. + Algorithm printExpression(v) 2 8 public void postOrderTraversal(Node n) if hasLeft (v) print(“(’’) { 1 4 79 if (n == null) printExpression (left(v)) ‐ return; print(v.element ()) 2 3 b 3 5 if hasRight (v) postOrderTraversal(n.left); printExpression (right(v)) a 1 postOrderTraversal(n.right); print (“)’’) System.out.print(n.value+" "); ((2 (a ‐ 1)) + (3 b)) } CSED233: Data Structures CSED233: Data Structures 7 by Prof. Bohyung Han, Fall 2013 8 by Prof. Bohyung Han, Fall 2013 Postorder Traversal Evaluate Arithmetic Expressions • Specialization of a postorder traversal 12 . Recursive method returning the value of a subtree . When visiting an internal node, combine the values of the subtrees 5 11 Algorithm evalExpr(v) 9 if isExternal (v) + 3 4 9 10 return v.element () 5 8 else 1 4 6 7 x evalExpr(leftChild (v)) 2 ‐ 3 2 1 2 8 y evalExpr(rightChild (v)) operator stored at v 2 3 5 1 return x y 6 7 2 5 1 ‐ x 3 2 x + CSED233: Data Structures CSED233: Data Structures 9 by Prof. Bohyung Han, Fall 2013 10 by Prof. Bohyung Han, Fall 2013 Binary Search Trees Search • A binary search tree is a binary tree storing keys at its nodes • Searching a key Algorithm TreeSearch(k, v) and satisfying the following property: . To search for a key k, we trace if T.isNull (v) . Let u, v, and w be three nodes such that u is the left child v and w is th a downward path starting at t return null e right child of v. Then, we have key(u) key(v) key(w). he root if k < key(v) return TreeSearch(k, T.left(v)) . The key value in v is larger than all keys in its left subtree and smaller . The next node visited depends than all keys in its right subtree. else if k = key(v) on the comparison of k with return v • An inorder traversal of a binary search trees visits the keys in the key of the current node increasing order. else { k > key(v) } . Search until we reach a leaf. return TreeSearch(k, T.right(v)) 6 • Example: get(4): . Call TreeSearch(4, root) 6 2 9 • The algorithms for floorEntry 2 9 1 4 8 and ceilingEntry are similar 1 4 8 CSED233: Data Structures CSED233: Data Structures 11 by Prof. Bohyung Han, Fall 2013 12 by Prof. Bohyung Han, Fall 2013 Insertion Insertion public void insert(int i) • Insertion at leaf 6 { . To perform operation 2 9 root = recursiveInsert(root,i); insert(k, o), we search for key } k using TreeSearch 1 4 8 . Assume k is not already in the private Node recursiveInsert(Node n, int i) tree, insert k by creating w as { one of the last visited node. w if (n == null) return new Node(i); • Example: insert 5 6 if (i<n.value) { // insert in the left subtree 2 9 n.left = recursiveInsert(n.left,i); return n; 1 4 8 } else { // insert in the right subtree w 5 n.right = recursiveInsert(n.right,i); return n; } } CSED233: Data Structures CSED233: Data Structures 13 by Prof. Bohyung Han, Fall 2013 14 by Prof. Bohyung Han, Fall 2013 Deletion Deletion (cont.) • Deleting a node with a child • Deleting a node with two 1 v . To perform operation remove 6 children 3 (k), we search for key k . 2 9 We consider the case where 2 8 . Assume key k is in the tree, the key k to be removed is and let let v be the node stori 1 4 v 8 stored at a node v whose 6 9 ng k . w children are both internal. w 5 5 . Simply connect parent and chi . we find the internal node w z ld of v. that follows v in an inorder traversal. • Example: remove 4 1 6 . we copy key(w) into node v. v 5 . we remove node w. 2 9 2 8 • Example: remove 3 1 5 8 6 9 CSED233: Data Structures CSED233: Data Structures 15 by Prof. Bohyung Han, Fall 2013 16 by Prof. Bohyung Han, Fall 2013 Implementation of Deletion Implementation of Deletion public void remove(int i) else { // match { if (n.left == null && n.right == null) { // no child root = recursiveRemove(root,i); return null; } } else if (n.left != null && n.right == null) { // left child only private Node recursiveRemove(Node n, int i) return n.left; { } if (n == null) // end of tree (node not found) else if (n.left == null && n.right != null) { // right child only return null; return n.right; } if (i < n.value) { // recurse left else { // two children n.left = recursiveRemove(n.left,i); Node maxLeft = findMax(n.left); // find node to replace return n; recursiveRemove(n.left, maxLeft.value); // remove the node } maxLeft.left = n.left; // set children of replacement node else if (i > n.value) { // recurse right maxLeft.right = n.right; n.right = recursiveRemove(n.right,i); return maxLeft; // return replacement node (adds it to tree) return n; } } } } CSED233: Data Structures CSED233: Data Structures 17 by Prof. Bohyung Han, Fall 2013 18 by Prof. Bohyung Han, Fall 2013 Performance • Consider ordered set items im plemented by means of a binary search tree of height . the space used is ܱ ݊ . methods get, put and remove take ܱ ݄ time. • The height is in the worst case and in the best cas e • We want a balanced binary tree! CSED233: Data Structures 20 19 by Prof. Bohyung Han, Fall 2013.