Someone give a strange birthday present to Ivan. It is hedgehog — connected undirected graph in which one vertex has degree at least $3$ (we will call it center) and all other vertices has degree 1. Ivan thought that hedgehog is too boring and decided to make himself $k$-multihedgehog.

Let us define $k$-multihedgehog as follows: $1$-multihedgehog is hedgehog: it has one vertex of degree at least $3$ and some vertices of degree 1. For all $k \ge 2$, $k$-multihedgehog is $(k-1)$-multihedgehog in which the following changes has been made for each vertex $v$ with degree 1: let $u$ be its only neighbor; remove vertex $v$, create a new hedgehog with center at vertex $w$ and connect vertices $u$ and $w$ with an edge. New hedgehogs can differ from each other and the initial gift. 

Thereby $k$-multihedgehog is a tree. Ivan made $k$-multihedgehog but he is not sure that he did not make any mistakes. That is why he asked you to check if his tree is indeed $k$-multihedgehog.


-----Input-----

First line of input contains $2$ integers $n$, $k$ ($1 \le n \le 10^{5}$, $1 \le k \le 10^{9}$) — number of vertices and hedgehog parameter.

Next $n-1$ lines contains two integers $u$ $v$ ($1 \le u, \,\, v \le n; \,\, u \ne v$) — indices of vertices connected by edge.

It is guaranteed that given graph is a tree.


-----Output-----

Print "Yes" (without quotes), if given graph is $k$-multihedgehog, and "No" (without quotes) otherwise.


-----Examples-----
Input
14 2
1 4
2 4
3 4
4 13
10 5
11 5
12 5
14 5
5 13
6 7
8 6
13 6
9 6

Output
Yes

Input
3 1
1 3
2 3

Output
No



-----Note-----

2-multihedgehog from the first example looks like this:

[Image]

Its center is vertex $13$. Hedgehogs created on last step are: [4 (center), 1, 2, 3], [6 (center), 7, 8, 9], [5 (center), 10, 11, 12, 13].

Tree from second example is not a hedgehog because degree of center should be at least $3$.

Difficulty: competition

Starter code:

Solutions:
from collections import defaultdict

n, k = list(map(int, input().split()))

connections = defaultdict(set)

for _ in range(n-1):
	u, v = list(map(int, input().split()))
	connections[u].add(v)
	connections[v].add(u)

leafs = set()
for node in connections:
	if len(connections[node])==1:
		leafs.add(node)

steps = 0
is_correct = True
while is_correct and steps<=k:
	new_leafs = set()
	for x in leafs:
		if len(connections[x])>1:
			is_correct = False
			#print("Len of %d more than one"%x)
			break
		root = list(connections[x])[0]
		if len(connections[root])<4 and len(leafs)!=3:
			is_correct = False
			#print("x: %d Len of root %d less than three"%(x,root))
			#print(connections[root])
			break
	if not is_correct:
		break
	for x in leafs:
		root = list(connections[x])[0]
		new_leafs.add(root)
		connections[root].remove(x)
	leafs = new_leafs
	steps += 1
	if len(leafs)==1 and len(connections[list(leafs)[0]])==0:
		break

#print("steps is %d"%steps)
if is_correct and steps==k:
	print("Yes")
else:
	print('No')


n,k = list(map(int,input().split(" ")))
degrees = [0] * n
neighbors = [list() for x in range(n)]
for i in range(n-1):
	first,second = list(map(int,input().split(" ")))
	degrees[first-1] += 1
	degrees[second-1] += 1
	neighbors[first-1] += [second]
	neighbors[second-1] += [first]

# start at a leaf
curr = 0
for i in range(n):
	if degrees[i] == 1:
		curr = i+1
		break
if curr == 0 or len(neighbors[curr-1]) == 0:
	print("No")
	return
curr = neighbors[curr-1][0]


def check(prev,parent,curr,level,degrees,neighbors,k):
	#print("curr: ",curr)
	#print("level: ",level)
	if level == 0:
		return len(parent) == 1 and degrees[curr-1] == 1,[]
	checked = []
	for neighbor in neighbors[curr-1]:
		#print("neighbor: ",neighbor)
		#print("checked: ",checked)
		#print("parent: ",parent)
		if len(prev) != 0 and prev[0] == neighbor:
			checked += [neighbor]
			continue
		if len(parent) != 0 and parent[0] == neighbor:
			continue
		result,garbage = check([],[curr],neighbor,level-1,degrees,neighbors,k)
		if result:
			checked += [neighbor]
		else:
			#print("adding the parent")
			if len(parent) == 0:
				parent += [neighbor]
			else:
				return False,[]
	if len(checked) > 2 and len(parent) == 0 and level == k:
		#print("first check")
		return True,[]
	elif len(checked) > 2 and len(parent) == 1 and level != k:
		#print("second check")
		return True,parent
	else:
		#print("len(checked): ",len(checked))
		#print("len(parent): ",len(parent))
		#print("level: ",level)
		#print("the end fail statement")
		return False,[]

prev = []
parent = []
counter = 1
while(counter <= k):
	result,parent = check(prev,[],curr,counter,degrees,neighbors,k)
	if not(result):
		print("No")
		return
	if counter == k:
		print("Yes")
		return
	prev = [curr]
	curr = parent[0]
	counter += 1

		




import math
n,k=map(int,input().split())
edges=[]
for i in range(n-1):
    edges.append(tuple(map(int,input().split())))
degreelist=[]
for i in range(min(k+1,math.floor(math.log2(n))+10)):
    degreelist.append({})
degrees=degreelist[0]
for i in range(1,n+1):
    degrees[i]=0
for guy in edges:
    degrees[guy[0]]+=1
    degrees[guy[1]]+=1
small=[]
center=None
done=False
for i in range(k):
    if not done:
        small=[]
        for guy in degrees:
            if degrees[guy]==2:
                print("No")
                done=True
                break
            if degrees[guy]==3:
                small.append(guy)
                if center==None: 
                    center=guy
                elif center!=guy:
                    print("No")
                    done=True
                    break
            elif degrees[guy]>1:
                small.append(guy)
        degrees=degreelist[i+1]
        if center!=None and center not in small:
            if not done:
                print("No")
            done=True
            break
        elif len(small)==0:
            if not done:
                print("No")
            done=True
            break
        for guy in small:
            degrees[guy]=0
        for guy in edges:
            if guy[0] in degrees and guy[1] in degrees:
                degrees[guy[0]]+=1
                degrees[guy[1]]+=1
        for guy in degrees:
            if degrees[guy]>1 and degreelist[i][guy]!=degrees[guy]:
                if not done:
                    print("No")
                done=True
                break
    else:
        break
if not done:
    if len(degreelist[-1])==1:
        print("Yes")
    else:
        print("No")

from collections import deque
 
m, k = list(map(int, input().split()))
 
G = [set() for _ in range(m + 1)]
 
q, nq = deque(), deque()
 
for _ in range(m - 1):
    u, v = list(map(int, input().split()))
    G[u].add(v)
    G[v].add(u)
 
for u in range(1, m + 1):
    if len(G[u]) == 1:
        q.append(u)
 
step = 0
removed = 0
ok = True
 
while removed < m - 1:
    each = {}
    for u in q:
        nxt = G[u].pop()
        G[nxt].remove(u)
        each[nxt] = each.get(nxt, 0) + 1
        removed += 1
        if len(G[nxt]) == 0:
            break
        if len(G[nxt]) == 1:
            nq.append(nxt)
    if any(v < 3 for k,v in list(each.items())):
        ok = False
        break
    q, nq = nq, deque()
    step += 1
 
if ok and step == k and removed == m - 1:
    print('Yes')
else:
    print('No')