Binary Search Trees

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# Binary Search Trees - PowerPoint PPT Presentation

Binary Search Trees. A binary tree: No node has more than two child nodes (called child subtrees). Child subtrees must be differentiated, into: Left-child subtree Right-child subtree A search tree: For every node, p: All nodes in the left subtree are < p

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## PowerPoint Slideshow about 'Binary Search Trees' - ronald

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Presentation Transcript
Binary Search Trees
• A binary tree:
• No node has more than two child nodes (called child subtrees).
• Child subtrees must be differentiated, into:
• Left-child subtree
• Right-child subtree
• A search tree:
• For every node, p:
• All nodes in the left subtree are < p
• All nodes in the right subtree are > p
Binary Search Trees (cont)
• Searching for a value is in a tree of N nodes is:
• O(log N) if the tree is “balanced”
• O(N) if the tree is “unbalanced”
“Unbalanced” Binary Search Trees
• Below is a binary search tree that is NOT “balanced”
Properties of Binary Trees
• A binary tree is a full binary tree if and only if:
• Each non leaf node has exactly two child nodes
• All leaf nodes have identical path length
• It is called full since all possible node slots are occupied
Full Binary Trees
• A Full binary tree of height h will have how many leaves?
• A Full binary tree of height h will have how many nodes?
Complete Binary Trees
• A complete binary tree (of height h) satisfies the following conditions:
• Level 0 to h-1 represent a fullbinary tree of height h-1
• One or more nodes in level h-1 may have 0, or 1 child nodes
• If j,k are nodes in level h-1, then j has more child nodes than k if and only if j is to the left of k
Complete Binary Trees (cont)
• Given a set of N nodes, a complete binary tree of these nodes provides the maximum number of leaves with the minimal average path length (per node)
• The complete binary tree containing n nodes must have at least one path from root to leaf of length log n
Height-balanced Binary Tree
• A height-balanced binary tree is a binary tree such that:
• The left & right subtrees for any given node differ in height by no more than one
• Note: Each complete binary tree is a height-balanced binary tree