Path from root to includes all ancestors of . 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 is the sum of arr[tin[u]] for all 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.