Base Case - First Column

The first column is easy: up[v][0] is the parent of $v$. If $v$ is the root, there is no parent. Set up[root][0] = root (or $-1$, or null, depending on your implementation).

This way, jumping up from the root keeps you at the root. For all other nodes, up[v][0] = parent[v]. You already have this from the input or from running BFS/DFS. This column serves as the foundation. Every later column builds on it through the recurrence relation.