How does the Root Method Work in Quick-Union?

Question

I've recently been studying the Princeton Algorithms course on Coursera. The course begins by teaching an algorithm called Quick-Union to solve the Dynamic Connectivity Problem..

While I understand the purpose and implementation of the algorithm - I'm unable to understand how the root function works in the code provided below. I have provided the class and the main method for my implementation.

public class QuickUnionUF {

private int[] id;

public QuickUnionUF(int N) {
    id = new int[N];
    for(int i = 0; i < N; i++) {
        id[i] = i;
    }
}

public int root(int k) {
    while(k != id[k]) { k = id[k]; }
    return k;
}

public  boolean connected(int p, int q) {
    return root(p) == root(q);
}

public void union(int p, int q) {
    int i = root(p);
    int j = root(q);
    id[i] = j;
}

public void print() {

    System.out.println("Indicies: 0 1 2 3 4 5 6 7 8 9");
    System.out.print("Elements: ");
    for(int i = 0; i < id.length; i++) {
        System.out.print(id[i] + " ");
    }
}


//Main Method 
public static void main(String[] args) {

    QuickUnionUF array = new QuickUnionUF(10);

    array.union(2, 9); 
    array.union(3, 4);
    array.union(4, 9);
    array.union(5, 6);

    System.out.println("\n The root of 3 is " + array.root(3));

}

//The union commands produce the following array: {0 1 9 4 9 6 6 7 8 9}

Output for the root of 3 The root of 3 is 9

I've arranged my array in such a way that it matches the array that the instructor is using, which is why I did those specific union commands. However, I cannot possibly understand why the root of 3 returns 9. Graphically, it makes sense, but the root method simply expresses:

while the number passed as a parameter doesn't equal the element it references, change the number entered (the index) to the element it references.

I believe the root method does this once for root(3). It seems to be only 1 iteration and simply returns 9. How did this happen? Shouldn't k (3) be set to id[k] (4) and then simply return 4 since now k = id[k]? It returns 9.. I used the debugger for some insight but couldn't come to a sound conclusion. There was another StackOverflow post on the exact question, but I didn't find that particularly insightful.