Problem - Fixed-Length Paths II

You are given a tree with $n$ nodes and integers $k_1$ and $k_2$. Count paths with length between $k_1$ and $k_2$ inclusive. This is a generalization of the previous problem. This is a generalization of Fixed-Length Paths I.

Instead of exactly $k$ edges, you count paths with $k_1 \le \text{length} \le k_2$. The centroid decomposition structure remains the same. The same centroid decomposition approach works with a small modification to the counting step. You will count ranges instead of exact values.