Binary Search TreeBinary Search Tree

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OutlineOutlineConcept of Binary Search Tree (BST)BST operations

◦Find◦Insert◦Remove

Running time analysis of BST operations

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Binary Search Tree: Binary Search Tree: PropertiesPropertiesElements have keys (no duplicates allowed).

For every node X in the tree, the values of all the keys in the left subtree are smaller than the key in X and the values of all the keys in the right subtree are larger than the key in X.

The keys must be comparable.

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<X >X

Binary Search Tree: Binary Search Tree: ExamplesExamples

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Binary Search Tree: Binary Search Tree: ExamplesExamples

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Basic Operations Basic Operations FindMin, FindMax, FindInsertRemove

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FindMinFindMinFind node with the smallest valueAlgorithm:

◦Keep going left until you reach a dead end!

Code:BinaryNode<Type> findMin(BinaryNode<Type> t) { if (t != null) while (t.left != null) t = t.left;

return t;}

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FindMaxFindMaxFind node with the largest valueAlgorithm:

◦Keep going right until you reach a dead end!

Code:BinaryNode<Type> findMax(BinaryNode<Type> t) { if (t != null) while (t.right != null) t = t.right;

return t;}

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FindFindYou are given an element to find in a BST. If it exists, return the node. If not, return null.

Algorithm?Code?

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Find: ImplementationFind: ImplementationBinaryNode<Type> find(Type x, BinaryNode<T> t) { while(t!=null) { if(x.compareTo(t.element)<0) t = t.left; else if(x.compareTo(t.element)>0) t = t.right; else return t; // Match }

return null; // Not found}

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Insertion: PrincipleInsertion: PrincipleWhen inserting a new element into a binary

search tree, it will always become a leaf node.

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Insertion: AlgorithmInsertion: AlgorithmTo insert X into a binary search tree:◦Start from the root◦ If the value of X < the value of the root:

X should be inserted in the left sub-tree. ◦ If the value of X > the value of the root:

X should be inserted in the right sub-tree.Remember that a sub-tree is also a tree.We can implement this recursively!

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Insertion: ImplementationInsertion: ImplementationBinaryNode<Type> insert(Type x, BinaryNode<Type> t) { if (t == null) t = new BinaryNode<Type>(x); else if(x.compareTo(t.element)<0) t.left = insert (x, t.left); else if(x.compareTo(t.element)>0) t.right = insert (x, t.right); else throw new DuplicateItemException(x);

return t;}

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Removing An ElementRemoving An Element

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Removing An Element: Removing An Element: AlgorithmAlgorithmIf the node is a leaf, simply delete it.If the node has one child, adjust parent’s

child reference to bypass the node.If the node has two children:

◦ Replace the node’s element with the smallest element in the right subtree and then remove that node, or

◦ Replace the node’s element with the largest element in the left subtree and then remove that node

Introduces new sub-problems: ◦ removeMin:◦ Alternatively, removeMax

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Removing LeafRemoving Leaf

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Removing Node With 1 Removing Node With 1 ChildChild

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Removing Node With 1 Removing Node With 1 ChildChild

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removeMin removeMin BinaryNode<Type> removeMin(BinaryNode<Type> t) { if (t == null) throw new ItemNotFoundException(); else if (t.left != null) {

t.left = removeMin(t.left); return t; } else return t.right;}

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Removing Node With 2 Removing Node With 2 ChildrenChildren

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Removing Node With 2 Removing Node With 2 ChildrenChildren

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Removing Node With 2 Removing Node With 2 ChildrenChildren

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Removing Root Removing Root

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Remove Remove BinaryNode<Type> remove(Type x, BinaryNode<Type> t){ if (t == null) throw new ItemNotFoundException(); if (x.compareTo(t.element)<0) t.left = remove(x, t.left); else if(x.compareTo(t.element)>0) t.right = remove(x, t.right); else if (t.left!=null && t.right != null) { t.element = findMin(t.right).element; t.right = removeMin(t.right); } else { if(t.left!=null) t=t.left; else t=t.right; } return t;}

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Find Find kk-th element-th element

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Find Find kk-th element-th elementBinaryNode<Type> findKth(int k, BinaryNode<Type> t) { if (t == null) throw exception; int leftSize = (t.left != null) ? t.left.size : 0;

if (k <= leftSize ) return findKth (k, t.left); else if (k == leftSize + 1) return t; else return findKth ( k - leftSize - 1, t.right); }

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AnalysisAnalysisRunning time for:

◦Insert?◦Find min?◦Remove?◦Find?

Average case: O(log n)Worst case: O(n)

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SummarySummaryBinary Search Tree maintains the

order of the tree.Each node should be comparableAll operations take O(log n) -

average case, when the tree is equally balanced.

All operations will take O(n) - worst case, when the height of the tree equals the number of nodes.

Smt Genap 2011-2012