Question

I’ve got such a structure, is described as a “binomial tree”. Let’see a drawing:

Which is the best way to represent this in memory? Just to clarify, is not a simple binary tree since the node N4 is both the left child of N1 and the right child of N2, the same sharing happens for N7 and N8 and so on… I need a construction algorithm tha easily avoid to duplicates such nodes, but just referencing them.

UPDATE
Many of us does not agree with the “binomial tree deefinition” but this cames from finance ( expecially derivative pricing ) have a look here: http://http.developer.nvidia.com/GPUGems2/gpugems2_chapter45.html for example. So I used the “Domain acceted definition”.

Answer 1

You could generate the structure level by level. In each iteration, create one level of nodes, put them in an array, and connect the previous level to them. Something like this (C#):

Node GenerateStructure(int levels)
{
    Node root = null;

    Node[] previous = null;

    for (int level = 1; level <= levels; level++)
    {
        int count = level;

        var current = new Node[count];

        for (int i = 0; i < count; i++)
            current[i] = new Node();

        if (level == 1)
            root = current[0];

        for (int i = 0; i < count - 1; i++)
        {
            previous[i].Left = current[i];
            previous[i].Right = current[i + 1];
        }

        previous = current;
    }

    return root;
}

The whole structure requires O(N^2) memory, where N is the number of level. This approach requires O(N) additional memory for the two arrays. Another approach would be to generate the graph from left to right, but that would require O(N) additional memory too.

The time complexity is obviously O(N^2).