Escape the Blue Chips

You are given a tree with $N$ vertices (numbered $1$ through $N$) rooted at vertex $1$. There are $M$ blue tokens (numbered $1$ through $M$) and $1$ red token. For each valid $i$, the $i$-th blue token appears at the root at an integer time $t_i$ seconds and moves to a leaf vertex $v_i$, crossing one edge per second. In the next second after reaching $v_i$, this token disappears. You control the