1338B - Edge Weight Assignment

题意：给出一棵树，要求给树上的每条边赋权值，使得任意两个叶子节点的路径上所有权值异或之后为0，边数n范围为[3,1e5]，求使用的不同权值数的最小值和最大值。

题解：

任取一个叶子节点为树根建树。则题意可转化为

首先考虑最小值

由异或的性质（a^b^b=a,总存在c使得对任意a,b有a^b^c=0）知：

当链长为2n+1的时候，我们总可以用唯一的数构造出一条路径使其与lca到根的值相等（设这个值为x，只需填充这条链的值全为x）

当链长为2n的时候

-如果lca到根的异或值为0，都填充1即可让这条链异或值为0

-否则：填充m组（1,2,3）,n组（1，1），使3m+2n+1=（链长），最后一个位置填充lca到根的值，不难看出这个值同样不大于3（因为依照前面的构造过程，不存在大于3的边，则异或结果也不大于3）

You have unweighted tree of $n$ vertices. You have to assign a positive weight to each edge so that the following condition would hold:

For every two different leaves $v_{1}$ and $v_{2}$ of this tree, bitwise XOR of weights of all edges on the simple path between $v_{1}$ and $v_{2}$ has to be equal to $0$ .

Note that you can put very large positive integers (like $10^{(10^{10})}$ ).

It's guaranteed that such assignment always exists under given constraints. Now let's define $f$ as the number of distinct weights in assignment.

$\text{[math]}$ In this example, assignment is valid, because bitwise XOR of all edge weights between every pair of leaves is

0

f

value is

2

here, because there are

2

distinct edge weights(

4

and

5

$\text{[math]}$ In this example, assignment is invalid, because bitwise XOR of all edge weights between vertex $1$ and vertex $6$ ( $3, 4, 5, 4$ ) is not $0$ .

What are the minimum and the maximum possible values of $f$ for the given tree? Find and print both.

Input

The first line contains integer $n$ ( $3 \leq n \leq 10^{5}$ ) — the number of vertices in given tree.

The $i$ -th of the next $n - 1$ lines contains two integers $a_{i}$ and $b_{i}$ ( $1 \leq a_{i} < b_{i} \leq n$ ) — it means there is an edge between $a_{i}$ and $b_{i}$ . It is guaranteed that given graph forms tree of $n$ vertices.

Output

Print two integers — the minimum and maximum possible value of $f$ can be made from valid assignment of given tree. Note that it's always possible to make an assignment under given constraints.

1338B - Edge Weight Assignment

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