Basic Union

To merge components containing $x$ and $y$, make one root point to the other:

function union(x, y):
    rootX = this.find(x)
    rootY = this.find(y)
    if rootX != rootY:
        this.parent[rootX] = rootY

After this, find(x) and find(y) return the same root.

Problem: without care, trees can become tall. If you union nodes $1, 2, 3, ..., n$ in order, you get a chain of length $n$. Then find takes $O(n)$ time.

You need two optimizations:

$1.$ Path compression: flatten trees during find

$2.$ Union by rank/size: keep trees balanced during union

Together, these achieve nearly constant time per operation.