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

Binary Search Tree - Example

Binary Search Trees (cont) Searching for a value is in a tree of N nodes is: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”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: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 occupiedIt is called full since all possible node slots are occupied

A Full Binary Tree - Example

Full Binary Trees A Full binary tree of height h will have how many leavesA Full binary tree of height h will have how many leaves? A Full binary tree of height h will have how many nodesA 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:A complete binary tree (of height h) satisfies the following conditions: –Level 0 to h-1 represent a full binary 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 - Example

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)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 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: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 Each complete binary tree is a height- balanced binary treeNote: Each complete binary tree is a height- balanced binary tree

Height-balanced Binary Tree - Example

Advantages of Height-balanced Binary Trees Height-balanced binary trees are “balanced”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 nodeslimited performance varianceOperations 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 networkVariance is a very important concern in real time applications, e.g. connecting calls in a telephone network