SG函数练习题：AGC017D - Game on Tree
https://atcoder.jp/contests/agc017/tasks/agc017_d
Problem Statement
There is a tree with \(N\) vertices numbered \(1, 2, ..., N\). The edges of the tree are denoted by \((x_i, y_i)\).
On this tree, Alice and Bob play a game against each other. Starting from Alice, they alternately perform the following operation:
- Select an existing edge and remove it from the tree, disconnecting it into two separate connected components. Then, remove the component that does not contain Vertex \(1\).
A player loses the game when he/she is unable to perform the operation. Determine the winner of the game assuming that both players play optimally.
Constraints
- \(2 \leq N \leq 100000\)
- \(1 \leq x_i, y_i \leq N\)
- The given graph is a tree.
Answer
经典SG函数题。
- 只有一个端点的树的SG函数为\(0\)。
- 根只有一个儿子的树中，其SG函数是其儿子子树的SG函数+1，数学归纳法即证。
- 根有多个儿子的树中，将所有根与儿子的连边拆出来形成成一个新的树（即，对所有边\((1,u)\)，将\(u\)的子树拆出来，再在\(u\)上连一个\((1_u,u)\)），则原树SG函数等于所有拆出来的数的SG函数的异或和。
