Problem:
Score : 400 points

Problem Statement

Given is a tree G with N vertices.
The vertices are numbered 1 through N, and the i-th edge connects Vertex a_i and Vertex b_i.

Consider painting the edges in G with some number of colors.
We want to paint them so that, for each vertex, the colors of the edges incident to that vertex are all different.

Among the colorings satisfying the condition above, construct one that uses the minimum number of colors.

Constraints

2 \le N \le 10^5

1 \le a_i \lt b_i \le N

All values in input are integers.

The given graph is a tree.

Input

Input is given from Standard Input in the following format:

N
a_1 b_1
a_2 b_2
\vdots
a_{N-1} b_{N-1}

Output

Print N lines.

The first line should contain K, the number of colors used.

The (i+1)-th line (1 \le i \le N-1) should contain c_i, the integer representing the color of the i-th edge, where 1 \le c_i \le K must hold.

If there are multiple colorings with the minimum number of colors that satisfy the condition, printing any of them will be accepted.

Sample Input 1

3
1 2
2 3

Sample Output 1

2
1
2

Sample Input 2

8
1 2
2 3
2 4
2 5
4 7
5 6
6 8

Sample Output 2

4
1
2
3
4
1
1
2

Sample Input 3

6
1 2
1 3
1 4
1 5
1 6

Sample Output 3

5
1
2
3
4
5

Solution in Rust (1.42.0) (judge verdict: Memory Limit Exceeded)

#![allow(unused_imports)]

// language: Rust(1.42.0)
// check available crates on AtCoder at "https://atcoder.jp/contests/language-test-202001"
// My Library Repositry is at "https://github.com/Loptall/sfcpl"

use itertools::*;
use itertools_num::*;
use lazy_static::lazy_static;
use maplit::{btreemap, btreeset, hashmap, hashset};
use num_bigint::{BigInt, BigUint};
use num_complex::Complex;
use num_integer::{binomial, gcd, lcm, multinomial, Integer};
use num_rational::Rational;
use num_traits::{
    clamp, one, pow, zero, Num, NumAssignOps, NumOps, One, Pow, Signed, Unsigned, Zero,
};
use permutohedron::Heap;
use proconio::{
    derive_readable, fastout, input, is_stdin_empty,
    marker::{Bytes, Chars, Isize1, Usize1},
};
use rand::random;
use std::cmp::{
    max, min, Ordering,
    Ordering::{Equal, Greater, Less},
    Reverse,
};
use std::collections::{BTreeMap, BTreeSet, BinaryHeap, HashMap, HashSet, VecDeque};
use std::convert::TryInto;
use std::fmt;
use std::mem::swap;
use std::num::{NonZeroU32, ParseIntError};
use std::ops::{
    Add, AddAssign, BitAnd, BitAndAssign, BitOr, BitOrAssign, BitXor, BitXorAssign, Div, DivAssign,
    Index, IndexMut, Mul, MulAssign, Neg, Rem, RemAssign, Shl, ShlAssign, Shr, ShrAssign, Sub,
    SubAssign,
};
use std::process::exit;
use std::{f32, f64, i128, i16, i32, i64, i8, isize, u128, u16, u32, u64, u8, usize};

pub use visualize::*;
pub mod visualize {
    use itertools::*;
    pub trait Visualize {
        fn visualize(&self, split: &str);
        fn continuous(&self) {
            self.visualize("");
        }
        fn spaces(&self) {
            self.visualize(" ");
        }
        fn lines(&self) {
            self.visualize("\n");
        }
    }
    macro_rules ! impl_vis_for_sized {($ ($ t : ty ) ,+ ) => {$ (impl Visualize for $ t {fn visualize (& self , _split : & str ) {print ! ("{}" , self ) ; } } ) + } ; }
    impl_vis_for_sized! {usize , u8 , u16 , u32 , u64 , u128 , isize , i8 , i16 , i32 , i64 , i128 , String , & str , char }
    impl<T: std::fmt::Display> Visualize for [T] {
        fn visualize(&self, split: &str) {
            print!("{}", self.iter().join(split));
        }
    }
    #[macro_export]
    macro_rules ! vis {() => {println ! () ; } ; ($ last : expr ; ) => {$ last . lines () ; vis ! () } ; ($ last : expr => ) => {$ last . continuous () ; vis ! () ; } ; ($ last : expr $ (, ) ? ) => {$ last . spaces () ; vis ! () ; } ; ($ first : expr ; $ ($ rest : tt ) * ) => {$ first . lines () ; println ! () ; vis ! ($ ($ rest ) * ) ; } ; ($ first : expr => $ ($ rest : tt ) * ) => {$ first . continuous () ; vis ! ($ ($ rest ) * ) ; } ; ($ first : expr , $ ($ rest : tt ) * ) => {$ first . spaces () ; print ! (" " ) ; vis ! ($ ($ rest ) * ) ; } ; }
}

pub use consts::*;
pub mod consts {
    pub const MOD10E9_7: usize = 1000000007; // 10 ^ 9 + 7
    pub const MOD99_: usize = 998244353;
    pub const MAX: usize = std::usize::MAX; // = 2 ^ 64 - 1 = 18446744073709551615 ≈ 1.8 * 10 ^ 19
    pub const INF: usize = 2000000000000000000; // MAX / 9 < 2 * 10e18 < MAX / 10
    pub const FNI: i64 = -2000000000000000000; // == -(INF as i64)
    pub const PI: f64 = std::f64::consts::PI; // 3.141592653589793 -- 10 ^ -15
    pub const ASCII_A_LARGE: u8 = 65;
    pub const ASCII_A_SMALL: u8 = 97;
    pub const ASCII_0: u8 = 48;
    pub const ADJ4: &[(isize, isize); 4] = &[(1, 0), (0, 1), (-1, 0), (0, -1)];
    pub const ADJ8: &[(isize, isize); 8] = &[
        (1, 0),
        (1, 1),
        (0, 1),
        (-1, 1),
        (-1, 0),
        (-1, -1),
        (0, -1),
        (1, -1),
    ];
}

pub trait Graph<'a> {
    type NodeId: Copy;
    type Iter: Iterator<Item = Self::NodeId>;
    fn len(&self) -> usize;
    fn is_empty(&self) -> bool;
    fn index(&self, i: Self::NodeId) -> usize;
    fn neighbors(&'a self, i: Self::NodeId) -> Self::Iter;
}
/// 重みなしグラフ
pub struct UnweightedGraph(Vec<Vec<usize>>);
impl UnweightedGraph {
    pub fn new(graph: Vec<Vec<usize>>) -> Self {
        Self(graph)
    }
}
impl<'a> Graph<'a> for UnweightedGraph {
    type NodeId = usize;
    type Iter = std::iter::Cloned<std::slice::Iter<'a, Self::NodeId>>;
    fn len(&self) -> usize {
        self.0.len()
    }
    fn is_empty(&self) -> bool {
        self.len() == 0
    }
    fn index(&self, a: Self::NodeId) -> usize {
        a
    }
    fn neighbors(&'a self, a: Self::NodeId) -> Self::Iter {
        self.0[a].iter().cloned()
    }
}
impl<'a> UnweightedGraph {
    pub fn dfs(&'a self, start: usize) -> Dfs<'a, UnweightedGraph> {
        dfs(self, start)
    }
    pub fn bfs(&'a self, start: usize) -> Bfs<'a, UnweightedGraph> {
        bfs(self, start)
    }
}
/// 重みありグラフ
pub struct WeightedNodeGraph<W>(Vec<Vec<(usize, W)>>);
impl<W> WeightedNodeGraph<W> {
    pub fn new(graph: Vec<Vec<(usize, W)>>) -> Self {
        Self(graph)
    }
}
impl<'a, W> Graph<'a> for WeightedNodeGraph<W>
where
    W: std::marker::Copy + Clone + 'a,
{
    type NodeId = (usize, W);
    type Iter = std::iter::Cloned<std::slice::Iter<'a, Self::NodeId>>;
    fn len(&self) -> usize {
        self.0.len()
    }
    fn is_empty(&self) -> bool {
        self.len() == 0
    }
    fn index(&self, a: Self::NodeId) -> usize {
        a.0
    }
    fn neighbors(&'a self, a: Self::NodeId) -> Self::Iter {
        self.0[a.0].iter().cloned()
    }
}
/// 重みなし有向グラフ
pub fn make_directed_graph(n: usize, edges: &[(usize, usize)]) -> UnweightedGraph {
    let mut g = vec![vec![]; n];
    for &(u, v) in edges.iter() {
        g[u].push(v);
    }
    UnweightedGraph::new(g)
}
/// 重みなし無向グラフ
pub fn make_undirected_graph(n: usize, edges: &[(usize, usize)]) -> UnweightedGraph {
    let mut g = vec![vec![]; n];
    for &(u, v) in edges.iter() {
        g[u].push(v);
        g[v].push(u);
    }
    UnweightedGraph::new(g)
}
/// 重み付き有向グラフ
pub fn make_weighted_directed_graph<W: Clone>(
    n: usize,
    edges: &[(usize, usize, W)],
) -> WeightedNodeGraph<W> {
    let mut g = vec![vec![]; n];
    for &(u, v, ref w) in edges.iter() {
        g[u].push((v, w.clone()));
    }
    WeightedNodeGraph::new(g)
}
/// 重み付き無向グラフ
pub fn make_weighted_undirected_graph<W: Clone>(
    n: usize,
    edges: &[(usize, usize, W)],
) -> WeightedNodeGraph<W> {
    let mut g = vec![vec![]; n];
    for &(u, v, ref w) in edges.iter() {
        g[u].push((v, w.clone()));
        g[v].push((u, w.clone()));
    }
    WeightedNodeGraph::new(g)
}

pub struct Dfs<'a, G: Graph<'a>> {
    visited: Vec<bool>,
    s: Vec<(G::NodeId, Option<G::NodeId>)>,
    g: &'a G,
}
impl<'a, G: Graph<'a>> Iterator for Dfs<'a, G> {
    type Item = (G::NodeId, G::NodeId);
    fn next(&mut self) -> Option<Self::Item> {
        if let Some((u, prev)) = self.s.pop() {
            for v in self.g.neighbors(u) {
                if !self.visited[self.g.index(v)] {
                    self.visited[self.g.index(v)] = true;
                    self.s.push((v, Some(u)));
                }
            }
            if let Some(prev) = prev {
                Some((prev, u))
            } else {
                self.next()
            }
        } else {
            None
        }
    }
}
/// `start`からの深さ優先探索時の、
/// 頂点から頂点の接続をイテレートするIteratorを作る
/// ```rust
/// use sfcpl::graph::{util::make_undirected_graph, dfs::dfs};
/// let e = &[(0, 2), (0, 1), (1, 2), (2, 3), (3, 4), (3, 5), (1, 5)];
/// let g = make_undirected_graph(6, e);
/// for (f, t) in dfs(&g, 0) {
///     println!("{} -> {}", f, t);
/// }
/// ```
/// これは
/// `0 -> 1`
/// `1 -> 5`
/// `5 -> 3`
/// `3 -> 4`
/// `0 -> 2`
/// を表示する
pub fn dfs<'a, G: Graph<'a, NodeId = usize>>(g: &'a G, start: G::NodeId) -> Dfs<'a, G> {
    let n = g.len();
    let mut visited = vec![false; n];
    let mut s = Vec::new();
    visited[start] = true;
    s.push((start, None));
    Dfs { visited, s, g }
}

/// 重みなしグラフにおいて、幅優先探索を用いて任意の頂点から、各頂点への最短距離を求める
/// dist[i] = (startからiまでの最短距離)
pub fn dist_table<'a, G>(g: &'a G, start: G::NodeId) -> Vec<usize>
where
    G: Graph<'a, NodeId = usize>,
{
    let mut dist = vec![std::usize::MAX; g.len()];
    dist[start] = 0;
    for (f, t) in bfs(g, start) {
        dist[t] = dist[f] + 1;
    }
    dist
}
/// 重みなしグラフでの2頂点の最短距離を幅優先探索で求める
/// goalまでの経路が見つかった時点で探索を打ち切るので、
/// グラフのサイズの大きさに直接の影響を受けないことが期待される
pub fn shortest_path<'a, G: Graph<'a, NodeId = usize>>(
    g: &'a G,
    start: usize,
    goal: usize,
) -> usize {
    let mut dist = vec![std::usize::MAX; g.len()];
    dist[start] = 0;
    for (f, t) in bfs(g, start) {
        if t == goal {
            return dist[f] + 1;
        }
        dist[t] = dist[f] + 1;
    }
    std::usize::MAX
}
pub struct Bfs<'a, G: Graph<'a>> {
    visited: Vec<bool>,
    q: VecDeque<(G::NodeId, Option<G::NodeId>)>,
    g: &'a G,
}
impl<'a, G: Graph<'a>> Iterator for Bfs<'a, G> {
    type Item = (G::NodeId, G::NodeId);
    fn next(&mut self) -> Option<Self::Item> {
        if let Some((u, prev)) = self.q.pop_front() {
            for v in self.g.neighbors(u) {
                if !self.visited[self.g.index(v)] {
                    self.visited[self.g.index(v)] = true;
                    self.q.push_back((v, Some(u)));
                }
            }
            if let Some(prev) = prev {
                Some((prev, u))
            } else {
                self.next()
            }
        } else {
            None
        }
    }
}
/// `start`の頂点からの幅優先探索時の、
/// 頂点から頂点の接続をイテレートするIteratorを作る
/// ```rust
/// use sfcpl::graph::{util::make_undirected_graph, bfs::bfs};
/// let e = &[(0, 2), (0, 1), (1, 2), (2, 3), (3, 4), (3, 5), (1, 5)];
/// let g = make_undirected_graph(6, e);
/// for (f, t) in bfs(&g, 0) {
///     println!("{} -> {}", f, t);
/// }
/// ```
/// これは
/// `0 -> 2`
/// `0 -> 1`
/// `2 -> 3`
/// `1 -> 5`
/// `3 -> 4`
/// を表示する
pub fn bfs<'a, G>(g: &'a G, start: G::NodeId) -> Bfs<'a, G>
where
    G: Graph<'a, NodeId = usize>,
{
    let n = g.len();
    let mut visited = vec![false; n];
    let mut q = VecDeque::new();
    visited[start] = true;
    q.push_back((start, None));
    Bfs { visited, q, g }
}

// #[fastout]
fn main() {
    input! {
        n: usize,
        edge: [(Usize1, Usize1); n - 1]
    }

    let m = {
        let mut c = vec![0usize; n];
        for &(a, b) in edge.iter() {
            c[a] += 1;
            c[b] += 1;
        }
        c.into_iter().max().unwrap()
    };

    let tree = make_directed_graph(n, &edge);
    let mut color_node = vec![vec![false; m]; n];
    let mut color_edge = HashMap::new();

    for (a, b) in tree.dfs(0) {
        let new_color = color_node[a].iter().zip(color_node[b].iter()).map(|(x, y)| x | y).enumerate().find(|x| !x.1).unwrap().0;
        color_edge.insert((a, b), new_color);
        color_node[a][new_color] = true;
        color_node[b][new_color] = true;
    }

    vis!(m);
    for e in edge {
        vis!(color_edge[&e] + 1);
    }
}