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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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binary search trees
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
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
“Unbalanced” Binary Search Trees
  • Below is a binary search tree that is NOT “balanced”
properties of binary trees
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
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
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
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
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
advantages of height balanced binary trees
Advantages of Height-balanced Binary Trees
  • Height-balanced binary trees are “balanced”
  • Operations that run in time proportional to the height of the tree are O(log n), n the number of nodes with limited performance variance
  • Variance is a very important concern in real time applications, e.g. connecting calls in a telephone network