Cats and Tree

You are given a rooted tree with $N$ nodes labelled $1$ through $N$. For each non-root node $i$, let's denote the direct parent of this node by $p_i$. $N$ cats, also labelled $1$ through $N$, have vanished from Chef's home. The cats are frightened, so they want to climb the tree one by one, in the order $C_1, C_2, \ldots, C_N$. At any time, each node may contain at most one cat. When cat $c$ want