# Tight upper bound for forming an $n$ element Red-Black Tree from scratch

I learnt that in a order-statistic tree (augmented Red-Black Tree, in which each node $$x$$ contains an extra field denoting the number of nodes in the sub-tree rooted at $$x$$) finding the $$i$$ th order statistics can be done in $$O(lg(n))$$ time in the worst case. Now in case of an array representing the dynamic set of elements finding the $$i$$ th order statistic can be achieved in the $$O(n)$$ time in the worst case.[ where $$n$$ is the number of elements].

Now I felt like finding a tight upper bound for forming an $$n$$ element Red-Black Tree so that I could comment about which alternative is better : "maintain the set elements in an array and perform query in $$O(n)$$ time" or "maintaining the elements in a Red-Black Tree (formation of which takes $$O(f(n))$$ time say) and then perform query in $$O(lg(n))$$ time".

So a very rough analysis is as follows, inserting an element into an $$n$$ element Red-Black Tree takes $$O(lg(n))$$ time and there are $$n$$ elements to insert , so it takes $$O(nlg(n))$$ time. Now this analysis is quite loose as when there are only few elements in the Red-Black tree the height is quite less and so is the time to insert in the tree.

I tried to attempt a detailed analysis as follows (but failed however):

Let while trying to insert the $$j=i+1$$ th element the height of the tree is atmost $$2.lg(i+1)+1$$. For an appropriate $$c$$, the total running time,

$$T(n)\leq \sum_{j=1}^{n}c.(2.lg(i+1)+1)$$

$$=c.\sum_{i=0}^{n-1}(2.lg(i+1)+1)$$

$$=c.\left[\sum_{i=0}^{n-1}2.lg(i+1)+\sum_{i=0}^{n-1}1\right]$$

$$=2c\sum_{i=0}^{n-1}lg(i+1)+cn\tag1$$

Now

$$\sum_{i=0}^{n-1}lg(i+1)=lg(1)+lg(2)+lg(3)+…+lg(n)=lg(1.2.3….n)\tag2$$

Now $$\prod_{k=1}^{n}k\leq n^n, \text{which is a very loose upper bound}\tag 3$$

Using $$(3)$$ in $$(2)$$ and substituting the result in $$(1)$$ we have $$T(n)=O(nlg(n))$$ which is the same as the rough analysis…

Can I do anything better than $$(3)$$?

All the nodes referred to are the internal nodes in the Red-Black Tree.