We present a distributed algorithm to compute the node search number in trees. This algorithm extends the centralized algorithm proposed by Ellis \emph{et al.}~[EST94]. It can be executed in an asynchronous environment, requires an overall computation time of $O(n\log{n})$, and $n$ messages of $\log_3{n}+4$ bits each. The main contribution of this work lies in the data structure proposed to design our algorithm, called \emph{hierarchical decomposition}. This simple and flexible data structure is used for four operations: updating the node search number after addition or deletion of any tree-edges in a distributed fashion; computing it in a tree whose edges are added sequentially and in any order; computing other graph invariants such as the process number and the edge search number, by changing only initialization rules; extending our algorithms for trees and forests of unknown size (using messages of up to $2\log_3{n}+5$ bits).