Editing Binary search (section)

==== Unsuccessful searches ====
Unsuccessful searches can be represented by augmenting the tree with ''external nodes'', which forms an ''extended binary tree''. If an internal node, or a node present in the tree, has fewer than two child nodes, then additional child nodes, called external nodes, are added so that each internal node has two children. By doing so, an unsuccessful search can be represented as a path to an external node, whose parent is the single element that remains during the last iteration. An ''external path'' is a path from the root to an external node. The ''external path length'' is the sum of the lengths of all unique external paths. If there are <math>n</math> elements, which is a positive integer, and the external path length is <math>E(n)</math>, then the average number of iterations for an unsuccessful search <math>T'(n)=\frac{E(n)}{n+1}</math>, with the one iteration added to count the initial iteration. The external path length is divided by <math>n+1</math> instead of <math>n</math> because there are <math>n+1</math> external paths, representing the intervals between and outside the elements of the array.{{Sfn|Knuth|1998|loc=§6.2.1 ("Searching an ordered table"), subsection "Further analysis of binary search"}}

This problem can similarly be reduced to determining the minimum external path length of all binary trees with <math>n</math> nodes. For all binary trees, the external path length is equal to the internal path length plus <math>2n</math>.{{Sfn|Knuth|1997|loc=§2.3.4.5 ("Path length")}} Substituting the equation for <math>I(n)</math>:{{Sfn|Knuth|1998|loc=§6.2.1 ("Searching an ordered table"), subsection "Further analysis of binary search"}}

<math>
E(n) = I(n) + 2n = \left[(n + 1)\left \lfloor \log_2(n + 1) \right \rfloor - 2^{\left \lfloor \log_2(n+1) \right \rfloor + 1} + 2\right] + 2n = (n + 1) (\lfloor \log_2 (n) \rfloor + 2) - 2^{\lfloor \log_2 (n) \rfloor + 1}
</math>

Substituting the equation for <math>E(n)</math> into the equation for <math>T'(n)</math>, the average case for unsuccessful searches can be determined:{{Sfn|Knuth|1998|loc=§6.2.1 ("Searching an ordered table"), subsection "Further analysis of binary search"}}

<math>
T'(n) = \frac{(n + 1) (\lfloor \log_2 (n) \rfloor + 2) - 2^{\lfloor \log_2 (n) \rfloor + 1}}{(n+1)} = \lfloor \log_2 (n) \rfloor + 2 - 2^{\lfloor \log_2 (n) \rfloor + 1}/(n + 1)
</math>