Core Idea - Path as Prefix

Path from root to $v$ includes all ancestors of $v$. In Euler tour, these ancestors have tin[u] <= tin[v] and tout[u] >= tout[v]. If you store values at entry times, the path sum from root to $v$ is the sum of arr[tin[u]] for all $u$ on the path.

But ancestors are not contiguous in the flattened array. You need a different approach. Breaking paths into these ranges lets you use data structures designed for array operations.