Algorithm- Sum of distances between every two nodes of a Binary Search Tree in O(n)?

Question

The question is to find out sum of distances between every two nodes of BinarySearchTree given that every parent-child pair is separated by unit distance. It is to be calculated

Answer 1

If you mean O(n) per each insertion, then it can be done, assuming you do it after each and every insertion, starting with the root.

0- Record the current sum of the distances. Call it s1: O(1).
1- Insert the new node: O(n).
2- Perform a BFS, starting at this new node.
   For each new node you discover, record its distance to the start (new) node, as BFS always does: O(n).
   This gives you an array of the distances from the start node to all other nodes.
3- Sum these distances up. Call this s2: O(n).
4- New_sum = s1 + s2: O(1).

This algorithm is thus O(n).