Tree Traversal. Lecture12: Tree II. Preorder Traversal

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Ezra Mathews
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1 Tree Traersal (0F) Lecture: Tree II Process of isiting nodes in a tree systematically Some algorithms need to isit all nodes in a tree. Example: printing, counting nodes, etc. Implementation Can be done easily by recursion Order of isits does matter. Computers R Us Bohyung Han Sales Manufacturing R&D CSE, POSTECH bhhan@postech.ac.kr US International Laptops Desktops Europe Asia Canada Preorder Traersal Preorder Traersal A preorder traersal of the subtree rooted at node n: Recursiely perform a preorder traersal of the left child. Recursiely perform a preorder traersal of the right child. A preorder traersal starts at the root. public oid preordertraersal(node n) preordertraersal(n.left); preordertraersal(n.right); public oid preordertraersal(node n) preordertraersal(n.left); preordertraersal(n.right); 4 4

Inorder Traersal Inorder Traersal A preorder traersal of the subtree rooted at node n: Recursiely perform an inorder traersal of the left child.

2 Inorder Traersal Inorder Traersal A preorder traersal of the subtree rooted at node n: Recursiely perform an inorder traersal of the left child. Recursiely perform an inorder traersal of the right child. An iorder traersal starts at the root. public oid inordertraersal(node n) public oid inordertraersal(node n) inordertraersal(n.left); inordertraersal(n.right); inordertraersal(n.left); inordertraersal(n.right); Defined only in binary tree. Print Arithmetic Expressions Postorder Traersal Specialization of an inorder traersal print operand or operator hen isiting node print ( before traersing left subtree print ) after traersing right subtree Algorithm printexpression() if hasleft () print( ( ) printexpression (left()) print(.element ()) if hasright () printexpression (right()) print ( ) ) + ( ) ( ) ( ) 4 b ( ) a A preorder traersal of the subtree rooted at node n: Recursiely perform a postorder traersal of the left child. Recursiely perform a postorder traersal of the right child. A postorder traersal starts at the root. public oid postordertraersal(node n) postordertraersal(n.left); postordertraersal(n.right);

Postorder Traersal Ealuate Arithmetic Expressions public oid postordertraersal(node n) Specialization of a postorder traersal Recursie method returning the alue of a subtree When isiting an internal

3 Postorder Traersal Ealuate Arithmetic Expressions public oid postordertraersal(node n) Specialization of a postorder traersal Recursie method returning the alue of a subtree When isiting an internal node, combine the alues of the subtrees postordertraersal(n.left); postordertraersal(n.right); Algorithm ealexpr() if isexternal () return.element () else x ealexpr(leftchild ()) y ealexpr(rightchild ()) operator stored at return x y 4 x x Binary Search Trees Search A binary search tree is a binary tree storing keys at its nodes and satisfying the folloing property: Let u,, and be three nodes such that u is the left child and is the right child of. Then, e hae key(u) key() key(). The key alue in is larger than all keys in its left subtree and smaller than all keys in its right subtree. An inorder traersal of a binary search trees isits the keys in an increasing order. u Searching a key To search for a key k, e trace a donard path starting at the root The next node isited depends on the comparison of k ith the key of the current node Search until e reach a leaf. Example: get(4) Call TreeSearch(4, root) The algorithms for floorentry and ceilingentry are similar. Algorithm TreeSearch(k, ) if T.isExternal () return if k < key() return TreeSearch(k, T.left()) else if k = key() return else return TreeSearch(k, T.right())

4 Insertion Insertion Inserting a node To perform operation insert(k), e search for key k using TreeSearch Assume k is not already in the tree, and let be the leaf reached by the search We insert k at node and expand into an internal node Example: insert public oid insert(int i) root = recursieinsert(root, i); priate Node recursieinsert(node n, int i) return ne Node(i); if (i < n.alue) // insert in the left subtree n.left = recursieinsert(n.left,i); else // insert in the right subtree n.right = recursieinsert(n.right,i); 4 Deletion Deletion (cont.) Deleting a node ith a child To perform operation remoe(k), e search for key k Assume key k is in the tree, and let be the node storing k Simply connect parent and child of. Example: remoe 4 Deleting a node ith to children We consider the case here the key k to be remoed is stored at a node hose children are both internal. We find the internal node that follos in an inorder traersal. e copy key() into node. e remoe node. Example: remoe z

Implementation of Deletion public oid delete(int i) root = recursieremoe(root,i); priate Node recursieremoe(node n, int i) // end of tree (node not found) return null; if (i < n.

5 Implementation of Deletion public oid delete(int i) root = recursieremoe(root,i); priate Node recursieremoe(node n, int i) // end of tree (node not found) return null; if (i < n.alue) // recurse left n.left = recursieremoe(n.left,i); else if (i > n.alue) // recurse right n.right = recursieremoe(n.right,i); Implementation of Deletion else // match if (n.left == null && n.right == null) // no child return null; else if (n.left!= null && n.right == null) // left child only return n.left; else if (n.left == null && n.right!= null) // right child only return n.right; else // to children Node maxleft = findmax(n.left); // find node to replace recursieremoe(n.left, maxleft.alue); // remoe the node maxleft.left = n.left; // set children of replacement node maxleft.right = n.right; return maxleft; // return replacement node (adds it to tree) Performance Consider ordered set items implemented by means of a binary search tree of height The space used is. Methods search, insert and delete take time. The height in the orst case. log in the best case. We ant a balanced binary tree! 0

Presentation for use with the textbook Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, 2015

Presentation for use with the textbook Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, 2015 Presentation for use ith the textbook Algorithm Design and Applications, by M. T. Goodrich and R. Tamassia, Wiley, 0 Ch.03 Binary Search Trees Binary Search Binary search can perform nearest neighbor queries