Optimal search in trees

It is well known that the optimal solution for searching in a finite total order set is binary search. In binary search we divide the set into two `halves' by querying the middle element and continue the search on the suitable half. What is the equivalent of binary search when the set P is partially ordered? A query in this case is to a point x∈P, with two possible answers: `yes' indicates that the required element is `below' x or `no' if the element is not below x. We show that the problem of computing an optimal strategy for search in posets that are tree-like (or forests) is polynomial in the size of the tree and requires at most O(n⁴ log³ n) steps. Optimal solutions of such search problems are often needed in program testing and debugging, where a given program is represented as a tree and a bug should be found using a minimal set of queries. This type of search is also applicable in searching classified large tree-like databases (e.g., the Internet).