P3128 [USACO15DEC] Max Flow P

Farmer John has installed a new system of $N-1$ pipes to transport milk between the $N$ stalls in his barn ($2 \leq N \leq 50,000$), conveniently numbered $1 \ldots N$. Each pipe connects a pair of stalls, and all stalls are connected to each-other via paths of pipes. FJ is pumping milk between $K$ pairs of stalls ($1 \leq K \leq 100,000$). For the $i$th such pair, you are told two stalls $s_i$ and $t_i$, endpoints of a path along which milk is being pumped at a unit rate. FJ is concerned that some stalls might end up overwhelmed with all the milk being pumped through them, since a stall can serve as a waypoint along many of the $K$ paths along which milk is being pumped. Please help him determine the maximum amount of milk being pumped through any stall. If milk is being pumped along a path from $s_i$ to $t_i$, then it counts as being pumped through the endpoint stalls $s_i$ and $t_i$, as well as through every stall along the path between them.

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$2 \le N \le 5 \times 10^4,1 \le K \le 10^5$