In this paper we propose a new algorithm for finding the blocks (biconnected components) of an undirected graph. A serial implementation runs in \(O(n+m)\) time and space on a graph of n vertices and m edges. A parallel implementation runs in O(log n) time and \(O(n+m)\) space using \(O(n+m)\) processors on a concurrent-read, concurrent-write parallel RAM. An alternative implementation runs in \(O(n^ 2/p)\) time and \(O(n^ 2)\) space using any number \(p\leq n^ 2/\log^ 2n\) of processors, on a concurrent-read, exclusive-write parallel RAM. The last algorithm has optimal speedup, assuming an adjacency matrix representation of the input. A general algorithmic technique that simplifies and improves computation of various functions on trees is introduced. This technique typically requires O(log n) time using processors and O(n) space on an exclusive-read exclusive-write parallel RAM.