Dynamic Connectivity Solution

class DynamicForest:
    nodes: array of Node

    function link(u, v)
        makeRoot(nodes[u])
        nodes[u].pathParent := nodes[v]

    function cut(u, v)
        makeRoot(nodes[u])
        access(nodes[v])
        // u is left child of v now
        nodes[v].left.parent := null
        nodes[v].left := null

    function connected(u, v)
        return findRoot(nodes[u]) = findRoot(nodes[v])

All operations $O(\log n)$ amortized.

For competitive programming, implement the full splay tree with access/makeRoot carefully. Off-by-one errors in rotation and parent updates are common.