二叉树搜索失败

Question

我知道对于一次成功的搜索，使用二叉树在包含n个键的所有输入上的平均搜索时间在Big O（lg n）中，但是对于不成功的研究，该结果是否成立？

Answer 1

Strictly speaking you may not attain O(log n) even for a successful search if your binary tree is unbalanced.

For a balanced binary tree it's easy to see that the result will hold even if it's unsuccessful, as at any generic step you are allowed to rule out significant fraction (around a half) of the remaining tree.

Answer 2

Yes, the binary tree structure allows you to essentially throw away half the data set at each check.

Just think about how many of the nodes you actually have to visit in order to make that determination that it's not in there.

Example:

     5
     /\
    3  8
   /\  /\
  1 4  6 9

Say you want 2. Start at the root, go left, throwing away 8 and its children. Smaller than 3 so go left throwing away 4. Larger than 1 so it's not there.

In this case you visited only 5, 3, and 1. You can think of it as the same as inserting a new node.