9. 二叉搜索树

Tree insert: searchs for that element A[i]
Tree walk:

Theorem:

E[height of a randomly built binary search tree] = O(lg n)

Proof outline

1. Prove Jensen's inequality:
   for convex function f
2. Instead of analyzing = the random variable of the height of a randomly constructed BST on n nodes, we analize
3. Prove that
4. Conclude that
   
   

(by Luke Devroy, 1986)

Proof 1.Jensen's inequality

is convex if for all and all with ,

vector and convex function

Lemma

if is convex & & & ,
then

灰色区域某一点意味着不等号右边

Induction
Base Case: When n = 1,

, 不等式

成立。
Induction Step:

Jensen's inequality

(f is a convex and X is a integer random variable)
proof:

Proof 2. Analysis Y_n

X_n = the random variable of the height of a randomly constructed BST on n nodes. Y_n = 2 ^X_n

If root r has rank k, then Xn = 1 + max{X_k-1, X_n-k}, Y_n = 2 * max{Y_k-1, Y_n-k}

define indicator r.v.'s

Pr{Z_nk = 1} = E[Z_nk] = 1/n

Proof 3. Analysis E[Y_n]

Then, we are going to use substitution.
Claim:
Proof:
The base case must be true when c is sufficiently large.
Induction Step:

Proof 4.

（简单地使用了Jensen's inequality）
( lg(O(n³)) = lg(cn³) = 3lg(n) + lg(c) )