Minimizing height of binary search tree

[s3a]

Homework Statement

Given a binary search tree that contains n nodes with keys 1, 2, 3, ... , n.

In what order should the keys be inserted into the binary search tree to obtain a tree with minimal height?

Homework Equations

Binary search tree algorithm.

The Attempt at a Solution

I'm assuming height and depth are the same in this case since I read that height is calculated by traversing from a leaf to a node whereas the depth is calculated by traversing from the root to a node. I believe that I must have each branch branch off to another two branches except for the leaves in order to minimize the height/depth. I know how a binary search tree works but given that we are dealing with an infinite amount of nodes, I don't know how to answer the question.

Any help would be greatly appreciated!
Thanks in advance!

 

[KataKoniK]

Hello! I can provide some insights on how to approach this problem. The order in which the keys are inserted into the binary search tree can greatly affect the height of the tree. In order to minimize the height, we need to ensure that the tree is balanced, meaning that the left and right subtrees of each node have almost equal number of nodes.

To achieve this, we can use a technique called "balanced insertion". This involves inserting the keys in a specific order that ensures the tree remains balanced at every step. The following steps can be followed:

1. Start by inserting the middle key (n/2) into the tree. This will be the root node.
2. Then, insert the middle key of the left subarray (1 to n/2 - 1) into the left subtree of the root.
3. Similarly, insert the middle key of the right subarray (n/2 + 1 to n) into the right subtree of the root.
4. Repeat this process recursively for each subarray until all the keys are inserted.

This approach ensures that the tree remains balanced at every step, resulting in a minimal height. In terms of code, this can be implemented using a modified version of the binary search tree algorithm. I hope this helps! Let me know if you have any further questions.