Finding the lightest simple path in trees with integer weights A tree with integer weights (positive, negative or zero) is given. We want to design an efficient algorithm for finding a simple path with lightest weight in this tree. That is, we look for shortest paths where the length of the path is equal to sum of all edges weights.

I know that in a tree there is only one simple path between every pair of nodes.
So, even if you try every pair of nodes finding the path between then, you would have an $O(n^3)$ algorithm.

A better way is to find for every node the cost to every other node in a single visit. This lowers the algorithm to $O(n^2)$.

Is there any algorithm that runs in linear time?
Order the tree, so we can speak about a node and its subtrees. Now evaluate the tree recursively. In each node keep track of two values: (1) the value of the optimal path below the node (starting and ending somewhere below in the subtrees), and (2) the value of the optimal path below that additionally ends in the node itself.