What is the Complexity of N-ary tree insertion and searching?

Question

I am implementing an N-1ry tree in C#. I am wondering how can I calculate the complexity of below methods. Here is my code:

Structure:

public class Node
{
    public int Value { get; set; }
    public Node Children { get; set; }
    public Node Sibilings { get; set; }

}

This method for searching:

public Node search(Node root, int data)
{
    if (root == null)
        return null;

    if (data == root.Value)
        return root;

    Node t = search(root.Children, data);
    if (t == null)
        t = search(root.Sibilings, data);
    return t;
}

This method for insertion:

public void Add(int[] data)
{
    Node temp = null;
    temp = search(ROOT, data[0]);

    if (temp == null)
        temp = new Node(data[0]);

    if (this.ROOT == null)
        ROOT = temp;
    Node parent = temp;

    for (int j = 1; j <= this.NoOfChildrens; j++)
    {
        // for first child
        if (j == 1)
        {
            parent.Children = new Node(data[j]);
            parent = parent.Children;
        }
        //for all other childs
        else
        {
            parent.Sibilings = new Node(data[j]);
            parent = parent.Sibilings;
        }

    }

}

Program entry point:

static void Main(string[] args)
{
    NAryTree naryTree = new NAryTree(3);

    // 1st element in each row is node Value,>=2nd....=>value of child

    int[][] data = { new int[] { 1, 2, 3, 4 }, new int[] { 2, 1, 6,0 }, new int[] { 3, 8, 9, 10 }, new int[] { 4, 0, 0, 0 } };

    naryTree.Add(data[0]);
    naryTree.Add(data[1]);
    naryTree.Add(data[2]);
    naryTree.Add(data[3]);
    naryTree.Add(new int[] {10,3,6,4});

    naryTree.preorder(naryTree.ROOT);
    Console.ReadLine();
}

What is the bigO complexity of these methods?

Answer 1

Let's see what we have in Search method. It is not a binary tree and we have recursion. So the Search method will call N times till we find a necessary value. So we can conclude that we have O(N) where N is the maximum(worst) number of iteration to find a value at the last iteration:

public Node search(Node root, int data)
{
    if (root == null)
        return null;

    if (data == root.Value)
        return root;

    Node t = search(root.Children, data);
    if (t == null)
        t = search(root.Sibilings, data);
    return t;
} 

For Addition method is simpler as we have for statement and no nested loops. So we have O(N) for Addition method.

As Wisconsin university says:

So for loops for (i = 0; i < N; i++) { sequence of statements } The loop executes N times, so the sequence of statements also executes N times. Since we assume the statements are O(1), the total time for the for loop is N * O(1), which is O(N) overall.