Count Paths Through Each Vertex in a Tree

You are given a tree with n vertices and two integer constants L and R. The tree is undirected and connected. A path is defined as the unique simple path between any two distinct vertices. The length of a path is the number of edges it contains.

Your task is to compute, for each vertex u (1 ≤ u ≤ n), the number of paths whose length d satisfies \[ L \le d \le R \] and that pass through vertex u (that is, vertex u lies on the unique simple path connecting the two endpoints, including the case where u is one of the endpoints).

Note: Each path is considered only once (order of endpoints does not matter).

inputFormat

The first line contains three integers: n L R, where n is the number of vertices, and L and R are the bounds for the path lengths.

Each of the next n-1 lines contains two integers u and v (1-based index) indicating that there is an edge between vertices u and v.

outputFormat

Output a single line containing n integers. The i-th integer is the number of paths (with length between L and R inclusive) that pass through vertex i. The numbers should be separated by spaces.

sample

3 1 2
1 2
2 3

2 3 2