Path Maximum Query and Path Maximum Sum Query in a Tree

Let T be a node-weighted tree with n nodes, and let π(u, v) denote the path between two nodes u and v in T. We address two problems: (i) Path Maximum Query: Preprocess T so that, for a query pair of nodes u and v, the maximum weight on π(u, v) can be found quickly. (ii) Path Maximum Sum Query: Preprocess T so that, for a query pair of nodes u and v, the maximum weight sum subpath of π(u, v) can be found quickly. For the problems we present solutions with O(1) query time and O(n log n) preprocessing time.