In what order should you insert a set of known keys into a B-Tree to get minimal height?

Question

Given a fixed number of keys or values(stored either in array or in some data structure) and order of b-tree, can we determine the sequence of inserting keys that would gene

Answer 1

To build a particular B-tree using Insert() as a black box, work backward. Given a nonempty B-tree, find a node with more than the minimum number of children that's as close to the leaves as possible. The root is considered to have minimum 0, so a node with the minimum number of children always exists. Delete a value from this node to be prepended to the list of Insert() calls. Work toward the leaves, merging subtrees.

For example, given the 2-3 tree

       8
   4       c
 2   6   a   e
1 3 5 7 9 b d f,

we choose 8 and do merges to obtain the predecessor

   4      c
 2   6  a   e
1 3 5 79 b d f.

Then we choose 9.

   4     c
 2   6 a   e
1 3 5 7 b d f

Then a.

    4    c
 2    6    e
1 3  5 7b d f

Then b.

   4   c
 2   6   e
1 3 5 7 d f

Then c.

   4
 2   6  e
1 3 5 7d f

Et cetera.