Buggy python submission (verdict: Time Limit Exceeded):
import sys
read = sys.stdin.buffer.read
input = sys.stdin.readline
input = sys.stdin.buffer.readline


sys.setrecursionlimit(10**9)
from functools import lru_cache

def RD(): return sys.stdin.read()
def II(): return int(input())
def MI(): return map(int,input().split())
def MF(): return map(float,input().split())
def LI(): return list(map(int,input().split()))
def LF(): return list(map(float,input().split()))
def TI(): return tuple(map(int,input().split()))
# rstrip().decode('utf-8')

#

def main():
	mod=10**9+7
	n=II()
	G=[[] for _ in range(n+1)]
	p=[0]*(n+1)
	v=[0]*(n+1)


	for _ in range(n-1):
		a,b=MI()
		G[a].append(b)
		G[b].append(a)

	inv2=[1]*(n+1)
	t=pow(pow(2,n,mod),mod-2,mod)
	for i in range(n,-1,-1):
		inv2[i]=t
		t*=2
		t%=mod
	#print(inv2)
	
	@lru_cache(maxsize=None)
	def dfs(a):
		cnt=1
		for i in G[a]:
			if p[a]==i:
				continue
			else:
				p[i]=a
				cnt+=dfs(i)
		v[a]=cnt
		return cnt
	dfs(1)
	#print(v)

	def f(a):
		return (1-inv2[a])*(1-inv2[n-a])%mod

	ans=1-inv2[n]-n*inv2[1]
	for i in range(1,n+1):
		ans+=f(v[i])%mod
	print(ans%mod)




if __name__ == "__main__":
	main()

Fixed submission:
import sys
read = sys.stdin.buffer.read
input = sys.stdin.readline
input = sys.stdin.buffer.readline


sys.setrecursionlimit(10**9)
from functools import lru_cache

def RD(): return sys.stdin.read()
def II(): return int(input())
def MI(): return map(int,input().split())
def MF(): return map(float,input().split())
def LI(): return list(map(int,input().split()))
def LF(): return list(map(float,input().split()))
def TI(): return tuple(map(int,input().split()))
# rstrip().decode('utf-8')

#

def main():
	mod=10**9+7
	n=II()
	G=[[] for _ in range(n+1)]
	p=[0]*(n+1)
	v=[0]*(n+1)


	for _ in range(n-1):
		a,b=MI()
		G[a].append(b)
		G[b].append(a)

	inv2=[1]*(n+1)
	t=pow(pow(2,n,mod),mod-2,mod)
	for i in range(n,-1,-1):
		inv2[i]=t
		t*=2
		t%=mod
	#print(inv2)

	def dfs(a):
		cnt=1
		for i in G[a]:
			if p[a]==i:
				continue
			else:
				p[i]=a
				cnt+=dfs(i)
		v[a]=cnt
		return cnt
	dfs(1)
	#print(v)

	def f(a):
		return (1-inv2[a])*(1-inv2[n-a])%mod

	ans=1-inv2[n]-n*inv2[1]
	for i in range(1,n+1):
		ans+=f(v[i])%mod
	print(ans%mod)



if __name__ == "__main__":
	main()

Buggy python submission (verdict: Time Limit Exceeded):
import sys
read = sys.stdin.buffer.read
input = sys.stdin.readline
input = sys.stdin.buffer.readline


sys.setrecursionlimit(10**9)
from functools import lru_cache

def RD(): return sys.stdin.read()
def II(): return int(input())
def MI(): return map(int,input().split())
def MF(): return map(float,input().split())
def LI(): return list(map(int,input().split()))
def LF(): return list(map(float,input().split()))
def TI(): return tuple(map(int,input().split()))
# rstrip().decode('utf-8')

#

def main():
	mod=10**9+7
	n=II()
	G=[[] for _ in range(n+1)]
	p=[0]*(n+1)
	v=[0]*(n+1)


	for _ in range(n-1):
		a,b=MI()
		G[a].append(b)
		G[b].append(a)

	inv2=[1]*(n+1)
	t=pow(pow(2,n,mod),mod-2,mod)
	for i in range(n,-1,-1):
		inv2[i]=t
		t*=2
		t%=mod
	#print(inv2)

	def dfs(a):
		cnt=1
		for i in G[a]:
			if p[a]==i:
				continue
			else:
				p[i]=a
				cnt+=dfs(i)
		v[a]=cnt
		return cnt
	dfs(1)
	#print(v)

	def f(a):
		return (1-inv2[a])*(1-inv2[n-a])%mod

	ans=1-inv2[n]-n*inv2[1]
	for i in range(1,n+1):
		ans+=f(v[i])%mod
	print(ans%mod)


if __name__ == "__main__":
	main()

Fixed submission:
import sys
read = sys.stdin.buffer.read
input = sys.stdin.readline
input = sys.stdin.buffer.readline


sys.setrecursionlimit(10**9)
from functools import lru_cache

def RD(): return sys.stdin.read()
def II(): return int(input())
def MI(): return map(int,input().split())
def MF(): return map(float,input().split())
def LI(): return list(map(int,input().split()))
def LF(): return list(map(float,input().split()))
def TI(): return tuple(map(int,input().split()))
# rstrip().decode('utf-8')

#

def main():
	mod=10**9+7
	n=II()
	G=[[] for _ in range(n+1)]
	p=[0]*(n+1)
	v=[0]*(n+1)


	for _ in range(n-1):
		a,b=MI()
		G[a].append(b)
		G[b].append(a)

	inv2=[1]*(n+1)
	t=pow(pow(2,n,mod),mod-2,mod)
	for i in range(n,-1,-1):
		inv2[i]=t
		t*=2
		t%=mod
	#print(inv2)

	def dfs(a):
		cnt=1
		for i in G[a]:
			if p[a]==i:
				continue
			else:
				p[i]=a
				cnt+=dfs(i)
		v[a]=cnt
		return cnt
	dfs(1)
	#print(v)

	def f(a):
		return (1-inv2[a])*(1-inv2[n-a])%mod

	ans=1-inv2[n]-n*inv2[1]
	for i in range(1,n+1):
		ans+=f(v[i])%mod
	print(ans%mod)



if __name__ == "__main__":
	main()

Buggy python submission (verdict: Time Limit Exceeded):
import sys
read = sys.stdin.buffer.read
input = sys.stdin.readline
input = sys.stdin.buffer.readline


sys.setrecursionlimit(10**9)
from functools import lru_cache

def RD(): return sys.stdin.read()
def II(): return int(input())
def MI(): return map(int,input().split())
def MF(): return map(float,input().split())
def LI(): return list(map(int,input().split()))
def LF(): return list(map(float,input().split()))
def TI(): return tuple(map(int,input().split()))
# rstrip().decode('utf-8')

#

def main():
	mod=10**9+7
	n=II()
	G=[[] for _ in range(n+1)]
	p=[0]*(n+1)
	v=[0]*(n+1)


	for _ in range(n-1):
		a,b=MI()
		G[a].append(b)
		G[b].append(a)

	inv2=[1]*(n+1)
	t=pow(pow(2,n,mod),mod-2,mod)
	for i in range(n,-1,-1):
		inv2[i]=t
		t*=2
		t%=mod
	#print(inv2)

	@lru_cache(maxsize=None)
	def dfs(a):
		cnt=1
		for i in G[a]:
			if p[a]==i:
				continue
			else:
				p[i]=a
				cnt+=dfs(i)
		v[a]=cnt
		return cnt
	dfs(1)
	#print(v)

	def f(a):
		return (1-inv2[a])*(1-inv2[n-a])%mod

	ans=1-inv2[n]-n*inv2[1]
	for i in range(1,n+1):
		ans+=f(v[i])%mod
	print(ans%mod)



















if __name__ == "__main__":
	main()

Fixed submission:
import sys
read = sys.stdin.buffer.read
input = sys.stdin.readline
input = sys.stdin.buffer.readline


sys.setrecursionlimit(10**9)
from functools import lru_cache

def RD(): return sys.stdin.read()
def II(): return int(input())
def MI(): return map(int,input().split())
def MF(): return map(float,input().split())
def LI(): return list(map(int,input().split()))
def LF(): return list(map(float,input().split()))
def TI(): return tuple(map(int,input().split()))
# rstrip().decode('utf-8')

#

def main():
	mod=10**9+7
	n=II()
	G=[[] for _ in range(n+1)]
	p=[0]*(n+1)
	v=[0]*(n+1)


	for _ in range(n-1):
		a,b=MI()
		G[a].append(b)
		G[b].append(a)

	inv2=[1]*(n+1)
	t=pow(pow(2,n,mod),mod-2,mod)
	for i in range(n,-1,-1):
		inv2[i]=t
		t*=2
		t%=mod
	#print(inv2)

	def dfs(a):
		cnt=1
		for i in G[a]:
			if p[a]==i:
				continue
			else:
				p[i]=a
				cnt+=dfs(i)
		v[a]=cnt
		return cnt
	dfs(1)
	#print(v)

	def f(a):
		return (1-inv2[a])*(1-inv2[n-a])%mod

	ans=1-inv2[n]-n*inv2[1]
	for i in range(1,n+1):
		ans+=f(v[i])%mod
	print(ans%mod)



if __name__ == "__main__":
	main()

Buggy python submission (verdict: Runtime Error):
import sys
read = sys.stdin.buffer.read
input = sys.stdin.readline
input = sys.stdin.buffer.readline


sys.setrecursionlimit(10**9)
from functools import lru_cache

def RD(): return sys.stdin.read()
def II(): return int(input())
def MI(): return map(int,input().split())
def MF(): return map(float,input().split())
def LI(): return list(map(int,input().split()))
def LF(): return list(map(float,input().split()))
def TI(): return tuple(map(int,input().split()))
# rstrip().decode('utf-8')

#

def main():
	mod=10**9+7
	n=II()
	G=[[] for _ in range(n+1)]
	e=[]
	v=[0]*(n+1)
	p=[0]*(n+1)

	for _ in range(n-1):
		a,b=MI()
		G[a].append(b)
		G[b].append(a)
		e.append([a,b,0])

	@lru_cache(maxsize=None)
	def tree(a):
		cnt=1
		for i in G[a]:
			if p[a]==i:
				continue
			else:
				p[i]=a
				cnt+=tree(i)
		v[a]=cnt
		return cnt
	tree(1)
	#print(v)

	for i in range(n-1):
		#print(e[i])
		a=e[i][0]
		b=e[i][1]
		e[i][2]=min(v[a],v[b])
	#print(e)

	ex=0

	for _,_,c in e:
		ex+=(1-0.5**c)*(1-0.5**(n-c))
	ex+=1-0.5**n-n/2
	ex=int(ex*2**n+0.5)
	#print(ex)

	for i in range(1,n+1):
		if ex%2==0:
			ex//=2
		else:
			i-=1
			break
	y=ex
	x=2**(n-i)
	#print(y,x)

	y%=mod
	x%=mod

	ans=y*pow(x,mod-2,mod)%mod

	print(ans)





































if __name__ == "__main__":
	main()

Fixed submission:
import sys
read = sys.stdin.buffer.read
input = sys.stdin.readline
input = sys.stdin.buffer.readline


sys.setrecursionlimit(10**9)
from functools import lru_cache

def RD(): return sys.stdin.read()
def II(): return int(input())
def MI(): return map(int,input().split())
def MF(): return map(float,input().split())
def LI(): return list(map(int,input().split()))
def LF(): return list(map(float,input().split()))
def TI(): return tuple(map(int,input().split()))
# rstrip().decode('utf-8')

#

def main():
	mod=10**9+7
	n=II()
	G=[[] for _ in range(n+1)]
	p=[0]*(n+1)
	v=[0]*(n+1)


	for _ in range(n-1):
		a,b=MI()
		G[a].append(b)
		G[b].append(a)

	inv2=[1]*(n+1)
	t=pow(pow(2,n,mod),mod-2,mod)
	for i in range(n,-1,-1):
		inv2[i]=t
		t*=2
		t%=mod
	#print(inv2)

	def dfs(a):
		cnt=1
		for i in G[a]:
			if p[a]==i:
				continue
			else:
				p[i]=a
				cnt+=dfs(i)
		v[a]=cnt
		return cnt
	dfs(1)
	#print(v)

	def f(a):
		return (1-inv2[a])*(1-inv2[n-a])%mod

	ans=1-inv2[n]-n*inv2[1]
	for i in range(1,n+1):
		ans+=f(v[i])%mod
	print(ans%mod)



if __name__ == "__main__":
	main()

Buggy python submission (verdict: Runtime Error):
import sys
read = sys.stdin.buffer.read
input = sys.stdin.readline
input = sys.stdin.buffer.readline


sys.setrecursionlimit(10**9)
from functools import lru_cache

def RD(): return sys.stdin.read()
def II(): return int(input())
def MI(): return map(int,input().split())
def MF(): return map(float,input().split())
def LI(): return list(map(int,input().split()))
def LF(): return list(map(float,input().split()))
def TI(): return tuple(map(int,input().split()))
# rstrip().decode('utf-8')

#

def main():
	mod=10**9+7
	n=II()
	G=[[] for _ in range(n+1)]
	e=[]
	v=[0]*(n+1)
	p=[0]*(n+1)

	for _ in range(n-1):
		a,b=MI()
		G[a].append(b)
		G[b].append(a)
		e.append([a,b,0])

	@lru_cache(maxsize=None)
	def tree(a):
		cnt=1
		for i in G[a]:
			if p[a]==i:
				continue
			else:
				p[i]=a
				cnt+=tree(i)
		v[a]=cnt
		return cnt
	tree(1)
	#print(v)

	for i in range(n-1):
		#print(e[i])
		a=e[i][0]
		b=e[i][1]
		e[i][2]=min(v[a],v[b])
	#print(e)

	ex=0

	for _,_,c in e:
		ex+=(1-0.5**c)*(1-0.5**(n-c))
	ex+=1-0.5**n-n/2
	ex=int(ex*2**n+0.5)
	#print(ex)

	for i in range(1,n+1):
		if ex%2==0:
			ex//=2
		else:
			i-=1
			break
	y=ex
	x=2**(n-i)
	#print(y,x)

	y%=mod
	x%=mod

	ans=y*pow(x,mod-2,mod)%mod

	print(ans)


if __name__ == "__main__":
	main()

Fixed submission:
import sys
read = sys.stdin.buffer.read
input = sys.stdin.readline
input = sys.stdin.buffer.readline


sys.setrecursionlimit(10**9)
from functools import lru_cache

def RD(): return sys.stdin.read()
def II(): return int(input())
def MI(): return map(int,input().split())
def MF(): return map(float,input().split())
def LI(): return list(map(int,input().split()))
def LF(): return list(map(float,input().split()))
def TI(): return tuple(map(int,input().split()))
# rstrip().decode('utf-8')

#

def main():
	mod=10**9+7
	n=II()
	G=[[] for _ in range(n+1)]
	p=[0]*(n+1)
	v=[0]*(n+1)


	for _ in range(n-1):
		a,b=MI()
		G[a].append(b)
		G[b].append(a)

	inv2=[1]*(n+1)
	t=pow(pow(2,n,mod),mod-2,mod)
	for i in range(n,-1,-1):
		inv2[i]=t
		t*=2
		t%=mod
	#print(inv2)

	def dfs(a):
		cnt=1
		for i in G[a]:
			if p[a]==i:
				continue
			else:
				p[i]=a
				cnt+=dfs(i)
		v[a]=cnt
		return cnt
	dfs(1)
	#print(v)

	def f(a):
		return (1-inv2[a])*(1-inv2[n-a])%mod

	ans=1-inv2[n]-n*inv2[1]
	for i in range(1,n+1):
		ans+=f(v[i])%mod
	print(ans%mod)



if __name__ == "__main__":
	main()