Buggy python submission (verdict: Time Limit Exceeded):
# coding: utf-8
import sys
#from operator import itemgetter
sysread = sys.stdin.buffer.readline
read = sys.stdin.buffer.read
from heapq import heappop, heappush
from collections import defaultdict
sys.setrecursionlimit(10**7)
import math
#from itertools import product, accumulate, combinations, product
#import bisect
#import numpy as np
#from copy import deepcopy
#from collections import deque
#from decimal import Decimal
#from numba import jit

INF = 1 << 50
EPS = 1e-8
mod = 10 ** 9 + 7

def mapline(t = int):
    return map(t, sysread().split())
def mapread(t = int):
    return map(t, read().split())

def dfs(c, to, cols, max_counts, pre_col = 0):
    colors = set()
    colors.add(pre_col)
    for idx, n in to[c]:
        if cols[idx]:continue
        num = 1
        for i in range(num, max_counts+1):
            if not i in colors:
                num = i
                cols[idx] = i
                colors.add(i)
                dfs(n, to, cols, max_counts, i)
                break


def run():
    N = int(sysread())
    to = [[] for _ in range(N+1)]
    cols = [0] * (N-1)
    counts = [0 for i in range(N+1)]
    for i in range(N-1):
        a,b = mapline()
        to[a].append((i, b))
        to[b].append((i, a))
        counts[a] += 1
        counts[b] += 1

    max_counts = max(counts)
    print(max_counts)
    dfs(1, to, cols, max_counts)

    for c in cols:
        print(c)

    #print(a ^ b)
if __name__ == "__main__":
    run()


Fixed submission:
# coding: utf-8
import sys
#from operator import itemgetter
sysread = sys.stdin.buffer.readline
read = sys.stdin.buffer.read
from heapq import heappop, heappush
from collections import defaultdict
sys.setrecursionlimit(10**7)
import math
#from itertools import product, accumulate, combinations, product
#import bisect
#import numpy as np
#from copy import deepcopy
#from collections import deque
#from decimal import Decimal
#from numba import jit

INF = 1 << 50
EPS = 1e-8
mod = 10 ** 9 + 7

def mapline(t = int):
    return map(t, sysread().split())
def mapread(t = int):
    return map(t, read().split())

def dfs(c, to, cols, max_counts, pre_col = 0):
    colors = set()
    colors.add(pre_col)
    num = 1
    for idx, n in to[c]:
        if cols[idx]:continue
        for i in range(num, max_counts+1):
            if not i in colors:
                num = i
                cols[idx] = i
                colors.add(i)
                dfs(n, to, cols, max_counts, i)
                break


def run():
    N = int(sysread())
    to = [[] for _ in range(N+1)]
    cols = [0] * (N-1)
    counts = [0 for i in range(N+1)]
    for i in range(N-1):
        a,b = mapline()
        to[a].append((i, b))
        to[b].append((i, a))
        counts[a] += 1
        counts[b] += 1

    max_counts = max(counts)
    print(max_counts)
    dfs(1, to, cols, max_counts)

    for c in cols:
        print(c)

    #print(a ^ b)
if __name__ == "__main__":
    run()

Buggy python submission (verdict: Time Limit Exceeded):
# coding: utf-8
import sys
#from operator import itemgetter
sysread = sys.stdin.buffer.readline
read = sys.stdin.buffer.read
from heapq import heappop, heappush
from collections import defaultdict
sys.setrecursionlimit(10**7)
import math
#from itertools import product, accumulate, combinations, product
#import bisect
#import numpy as np
#from copy import deepcopy
#from collections import deque
#from decimal import Decimal
#from numba import jit

INF = 1 << 50
EPS = 1e-8
mod = 10 ** 9 + 7

def mapline(t = int):
    return map(t, sysread().split())
def mapread(t = int):
    return map(t, read().split())

def dfs(c, to, cols, max_counts, pre_col = 0):
    colors = set()
    colors.add(pre_col)
    for idx, n in to[c]:
        if cols[idx]:continue
        for i in range(1, max_counts+1):
            if not i in colors:
                cols[idx] = i
                colors.add(i)
                dfs(n, to, cols, max_counts, i)
                break


def run():
    N = int(sysread())
    to = [[] for _ in range(N+1)]
    cols = [0] * (N-1)
    counts = [0 for i in range(N+1)]
    for i in range(N-1):
        a,b = mapline()
        to[a].append((i, b))
        to[b].append((i, a))
        counts[a] += 1
        counts[b] += 1

    max_counts = max(counts)
    print(max_counts)
    dfs(1, to, cols, max_counts)

    for c in cols:
        print(c)




    #print(a ^ b)
if __name__ == "__main__":
    run()


Fixed submission:
# coding: utf-8
import sys
#from operator import itemgetter
sysread = sys.stdin.buffer.readline
read = sys.stdin.buffer.read
from heapq import heappop, heappush
from collections import defaultdict
sys.setrecursionlimit(10**7)
import math
#from itertools import product, accumulate, combinations, product
#import bisect
#import numpy as np
#from copy import deepcopy
#from collections import deque
#from decimal import Decimal
#from numba import jit

INF = 1 << 50
EPS = 1e-8
mod = 10 ** 9 + 7

def mapline(t = int):
    return map(t, sysread().split())
def mapread(t = int):
    return map(t, read().split())

def dfs(c, to, cols, max_counts, pre_col = 0):
    colors = set()
    colors.add(pre_col)
    num = 1
    for idx, n in to[c]:
        if cols[idx]:continue
        for i in range(num, max_counts+1):
            if not i in colors:
                num = i
                cols[idx] = i
                colors.add(i)
                dfs(n, to, cols, max_counts, i)
                break


def run():
    N = int(sysread())
    to = [[] for _ in range(N+1)]
    cols = [0] * (N-1)
    counts = [0 for i in range(N+1)]
    for i in range(N-1):
        a,b = mapline()
        to[a].append((i, b))
        to[b].append((i, a))
        counts[a] += 1
        counts[b] += 1

    max_counts = max(counts)
    print(max_counts)
    dfs(1, to, cols, max_counts)

    for c in cols:
        print(c)

    #print(a ^ b)
if __name__ == "__main__":
    run()