Problem Statement

There is a tree with $N$ vertices numbered $1$ to $N$. The $i$-th edge of the tree connects vertices $u_i$ and $v_i$ bidirectionally.

For a permutation $P=(P_1,\ldots,P_N)$ of $(1,\ldots,N)$, we define $f(P)$ as follows:

• For each vertex $i\ (1\leq i\leq N)$, let $(v_1=1,v_2,\ldots,v_k=i)$ be the simple path from vertex $1$ to vertex $i$, and let $L_i$ be the length of a longest increasing subsequence of $(P_{v_1},P_{v_2},\ldots,P_{v_k})$. We define $f(P) = \sum_{i=1}^N L_i$.

You are given an integer $K$. Determine whether there is a permutation $P$ of $(1,\ldots,N)$ such that $f(P)=K$. If it exists, present one such permutation.

What is a longest increasing subsequence? A subsequence of a sequence is a sequence obtained by removing zero or more elements from the original sequence and concatenating the remaining elements without changing the order. For example, $(10,30)$ is a subsequence of $(10,20,30)$, but $(20,10)$ is not. The longest increasing subsequence of a sequence is the longest strictly increasing subsequence of that sequence.

What is a simple path? For vertices $X$ and $Y$ in a graph $G$, a walk from vertex $X$ to vertex $Y$ is a sequence of vertices $(v_1,v_2, \ldots, v_k)$ such that $v_1=X$, $v_k=Y$, and $v_i$ and $v_{i+1}$ are connected by an edge for all $1\leq i\leq k-1$. Furthermore, if $v_1,v_2, \ldots, v_k$ are all distinct, it is called a simple path (or simply a path) from vertex $X$ to vertex $Y$.

Constraints

• All input values are integers.
• $2 \leq N \leq 2\times 10^5$
• $1\leq K \leq 10^{11}$
• $1\leq u_i,v_i\leq N$
• The given graph is a tree.

Input

The input is given from Standard Input in the following format:

N K
u_1 v_1
\vdots
u_{N-1} v_{N-1}

Output

If there is no permutation $P$ such that $f(P)=K$, print No.

If there is a permutation $P$ such that $f(P)=K$, print it in the following format:

Yes
P_1 \ldots P_N

If multiple permutations $P$ satisfy the condition, any of them will be accepted.