P2982 [USACO10FEB] Slowing down G

Every day, each of Farmer John's $N$ $(1 \le N \le 100{,}000)$ cows, conveniently numbered $1..N$, moves from the barn to her private pasture. The pastures form a tree, with the barn at pasture $1$. Exactly $N-1$ undirected paths connect the pastures; directly connected pastures have exactly one path. Path $i$ connects pastures $A_i$ and $B_i$ $(1 \le A_i \le N, 1 \le B_i \le N)$. Cow $i$ has a private pasture $P_i$ $(1 \le P_i \le N)$. The barn's small door lets only one cow exit at a time, and the patient cows wait until their predecessor arrives at her private pasture. First, cow $1$ exits and moves to pasture $P_1$. Then cow $2$ exits and goes to pasture $P_2$, and so on. While cow $i$ walks to $P_i$, she might or might not pass through a pasture that already contains an eating cow. When a cow is present in a pasture, cow $i$ walks slower than usual while passing through that pasture to avoid annoying her friend. ```cpp Consider the following pasture network, where the number between parentheses indicates the pastures' owner. 1 (3) / \ (1) 4 3 (5) / \ (2) 2 5 (4) First, cow 1 walks to her pasture: 1 (3) / \ [1] 4* 3 (5) / \ (2) 2 5 (4) When cow 2 moves to her pasture, she first passes into the barn's pasture, pasture 1. Then she sneaks around cow 1 in pasture 4 before arriving at her own pasture. 1 (3) / \ [1] 4* 3 (5) / \ [2] 2* 5 (4) Cow 3 doesn't get far at all -- she lounges in the barn's pasture, #1. 1* [3] / \ [1] 4* 3 (5) / \ [2] 2* 5 (4) Cow 4 must slow for pasture 1 and 4 on her way to pasture 5: 1* [3] / \ [1] 4* 3 (5) / \ [2] 2* 5* [4] Cow 5 slows for cow 3 in pasture 1 and then enters her own private pasture: 1* [3] / \ [1] 4* 3*[5] / \ [2] 2* 5* [4] ``` FJ would like to know how many times each cow has to slow down.

- Line 1: An integer $N$. - Lines $2..N$: Line $i+1$ contains two space-separated integers $A_i$ and $B_i$. - Lines $N+1..N+N$: Line $N+i$ contains a single integer $P_i$.

- Lines $1..N$: Line $i$ contains the number of times cow $i$ has to slow down.

Constraints: $1 \leq A_i, B_i, P_i \leq N \leq 10^5$. Translated by ChatGPT 5