Product of all nodes except for one in Binary Tree
One simple approach: for an internal node $x$, let $P(x)$ denote the product of the integers on the path from the root to $x$, and let $Q(x)$ denote the set of integers obtainable as the product of all but one of the integers on the path from the root to $x$.  Now, descend the tree from the root down to the leaves, computing $P(x)$ and $Q(x)$ for each node $x$ as you go.  Note that you can compute $P(x),Q(x)$ from $P(w),Q(w)$, where $w$ is the parent of $x$.

The asymptotic worst-case running time will be $O(n^2)$.  No algorithm has better worst-case asymptotic running time, as there can be $\Theta(n^2)$ different products that you need to output, so any correct algorithm will need to have worst-case running time at least $\Theta(N^2)$.