Tree Edge Weight Assignment

You are given a tree $G$ with $n$ nodes and an integer $E$. Your task is to assign an integer weight $w_i$ to each edge of the tree such that:

1. $1 \le w_i \le 10^9$ for every edge.
2. Define $dis(u,v)$ as the sum of the weights along the unique simple path between nodes $u$ and $v$. For a node $u$, let [ M(u)= \max_{v=1}^{n} dis(u,v), ] and consider a node $u$ chosen uniformly at random from $1$ to $n$. The expected value of $M(u)$, taken modulo $998244353$, must be exactly $E$.

If there is no valid assignment of weights that satisfies these conditions, output -1.

If you are not familiar with computing expectations and working with modular arithmetic in this setting, please refer to the template problem P2613 \textbf{Template: Rational Numbers Modulo} for guidance.

inputFormat

The first line contains two integers $n$ and $E$ ($1 \le n \le 10^5$ and $0 \le E \le 998244352$). The following $n-1$ lines each contain two integers $u$ and $v$ describing an edge between nodes $u$ and $v$ in the tree.

outputFormat

If there exists an assignment of weights satisfying the conditions, output $n-1$ space‐separated integers representing the weight for each edge in the order of input. Otherwise, output -1.

Note: In a tree with one node ($n=1$), since there are no edges, output an empty line if $E=0$, otherwise output -1.

sample

1 0