{"problem":{"name":"XOR Tree","description":{"content":"You are given a tree with $N$ vertices. The vertices are numbered $0$ through $N-1$, and the edges are numbered $1$ through $N-1$. Edge $i$ connects Vertex $x_i$ and $y_i$, and has a value $a_i$. You ","description_type":"Markdown"},"platform":"AtCoder","limit":{"time_limit":2000,"memory_limit":262144},"difficulty":"None","is_remote":true,"is_sync":true,"sync_url":null,"sign":"apc001_f"},"statements":[{"statement_type":"Markdown","content":"You are given a tree with $N$ vertices. The vertices are numbered $0$ through $N-1$, and the edges are numbered $1$ through $N-1$. Edge $i$ connects Vertex $x_i$ and $y_i$, and has a value $a_i$. You can perform the following operation any number of times:\n\n*   Choose a simple path and a non-negative integer $x$, then for each edge $e$ that belongs to the path, change $a_e$ by executing $a_e ← a_e ⊕ x$ (⊕ denotes XOR).\n\nYour objective is to have $a_e = 0$ for all edges $e$. Find the minimum number of operations required to achieve it.\n\n## Constraints\n\n*   $2 ≤ N ≤ 10^5$\n*   $0 ≤ x_i,y_i ≤ N-1$\n*   $0 ≤ a_i ≤ 15$\n*   The given graph is a tree.\n*   All input values are integers.\n\n## Input\n\nInput is given from Standard Input in the following format:\n\n$N$\n$x_1$ $y_1$ $a_1$\n$x_2$ $y_2$ $a_2$\n$:$\n$x_{N-1}$ $y_{N-1}$ $a_{N-1}$\n\n[samples]","is_translate":false,"language":"English"}],"meta":{"iden":"apc001_f","tags":[],"sample_group":[["5\n0 1 1\n0 2 3\n0 3 6\n3 4 4","3\n\nThe objective can be achieved in three operations, as follows:\n\n*   First, choose the path connecting Vertex $1, 2$, and $x = 1$.\n*   Then, choose the path connecting Vertex $2, 3$, and $x = 2$.\n*   Lastly, choose the path connecting Vertex $0, 4$, and $x = 4$."],["2\n1 0 0","0"]],"created_at":"2026-03-03 11:01:14"}}