Lecture12: Tree II Bohyung Han CSE, POSTECH [email protected] CSED233: Data Structures (2014F)

Lecture12: Tree II

Bohyung HanCSE, POSTECH

[email protected]

CSED233: Data Structures (2014F)

2 CSED233: Data Structuresby Prof. Bohyung Han, Fall 2014

Tree Traversal

• Process of visiting nodes in a tree systematically Some algorithms need to visit all nodes in a tree. Example: printing, counting nodes, etc.

• Implementation Can be done easily by recursion Order of visits does matter.

Computers”R”Us

Sales

R&D

Manufactur-ing

Lap-tops

Desk-tops

US

Interna-tional

Eu-rope

Asia

Canada


Preorder Traversal

• A preorder traversal of the subtree rooted at node n: Visit node n (process the node's data). Recursively perform a preorder traversal of the left child. Recursively perform a preorder traversal of the right child.

• A preorder traversal starts at the root.

public void preOrderTraversal(Node n){ if (n == null) return;

System.out.print(n.value+" "); preOrderTraversal(n.left); preOrderTraversal(n.right);}


Preorder Traversal

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public void preOrderTraversal(Node n){ if (n == null) return;

System.out.print(n.value+" "); preOrderTraversal(n.left); preOrderTraversal(n.right);}


Inorder Traversal

• A preorder traversal of the subtree rooted at node n: Recursively perform an inorder traversal of the left child. Visit node n (process the node's data). Recursively perform an inorder traversal of the right child.

• An iorder traversal starts at the root.

public void inOrderTraversal(Node n){ if (n == null) return;

inOrderTraversal(n.left); System.out.print(n.value+" "); inOrderTraversal(n.right);}

Defined only in binary tree.


Inorder Traversal

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public void inOrderTraversal(Node n){ if (n == null) return;

inOrderTraversal(n.left); System.out.print(n.value+" "); inOrderTraversal(n.right);}


Print Arithmetic Expressions

• Specialization of an inorder traversal print operand or operator when visiting node print “(“ before traversing left subtree print “)“ after traversing right subtree

Algorithm printExpression(v)if hasLeft (v)

print(“(’’)printExpression (left(v))

print(v.element ())if hasRight (v)

printExpression (right(v))print (“)’’)

+

÷

-2

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3 b3

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( (𝟐÷ (𝒂−𝟏 ) )+(𝟑×𝒃) )

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)


Postorder Traversal

• A preorder traversal of the subtree rooted at node n: Recursively perform a postorder traversal of the left child. Recursively perform a postorder traversal of the right child. Visit node n (process the node's data).

• A postorder traversal starts at the root.

public void postOrderTraversal(Node n){ if (n == null) return;

postOrderTraversal(n.left); postOrderTraversal(n.right); System.out.print(n.value+" ");}


Postorder Traversal

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public void postOrderTraversal(Node n){ if (n == null) return;

postOrderTraversal(n.left); postOrderTraversal(n.right); System.out.print(n.value+" ");}


Evaluate Arithmetic Expressions

• Specialization of a postorder traversal Recursive method returning the value of a subtree When visiting an internal node, combine the values of the subtrees

Algorithm evalExpr(v)if isExternal (v)

return v.element ()else

x evalExpr(leftChild (v))y evalExpr(rightChild

(v)) operator stored at v

return x y

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2 5 1 x 3 2 x +‐


Binary Search Trees

• A binary search tree is a binary tree storing keys at its nodesand satisfying the following property: Let u, v, and w be three nodes such that u is the left child v and w is

the right child of v. Then, we have key(u) key(v) key(w). The key value in v is larger than all keys in its left subtree and smaller

than all keys in its right subtree.

• An inorder traversal of a binary search trees visits the keys inan increasing order.

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u w


Search

• Searching a key To search for a key k, we trace

a downward path starting at the root

The next node visited depends on the comparison of k withthe key of the current node

Search until we reach a leaf.

• Example: get(4) Call TreeSearch(4, root)

• The algorithms for floorEn-tryand ceilingEntry are simi-lar.

Algorithm TreeSearch(k, v)if T.isExternal (v)

return vif k < key(v)

return TreeSearch(k, T.left(v))else if k = key(v)

return velse

return TreeSearch(k, T.right(v))

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Insertion

• Inserting a node To perform operation insert(k), we search for key k using TreeSearch

Assume k is not already in the tree, and let w be the leaf reached by the search

We insert k at node w and expand w into an internal node

• Example: insert 5

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w

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5w


Insertion

public void insert(int i){ root = recursiveInsert(root, i);}

private Node recursiveInsert(Node n, int i){ if (n == null) return new Node(i);

if (i < n.value) { // insert in the left subtree n.left = recursiveInsert(n.left,i); return n; } else { // insert in the right subtree n.right = recursiveInsert(n.right,i); return n; }}


Deletion

• Deleting a node with a child To perform operation remove(k), we search for key k

Assume key k is in the tree, and let v be the node storing k

Simply connect parent and child of v.

• Example: remove 4

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vw

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Deletion (cont.)

• Deleting a node with two children We consider the case where

the key k to be removed is stored at a node v whose children are both internal.

We find the internal node w that follows v in an inorder traversal.

we copy key(w) into node v. we remove node w.

• Example: remove 3

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Implementation of Deletion

public void delete(int i){ root = recursiveRemove(root,i);}

private Node recursiveRemove(Node n, int i){ if (n == null) // end of tree (node not found) return null;

if (i < n.value) { // recurse left n.left = recursiveRemove(n.left,i); return n; } else if (i > n.value) { // recurse right n.right = recursiveRemove(n.right,i); return n; }


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 { // two children Node maxLeft = findMax(n.left); // find node to replace recursiveRemove(n.left, maxLeft.value); // remove 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 worst case. in the best case.

• We want a balanced binary tree!

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Documents

Lecture12: Tree II Bohyung Han CSE, POSTECH [email protected] CSED233: Data Structures (2014F)