Several fast algorithms are presented for computing functions defined on paths in trees under various assumpuons. The algorithms are based on tree mampulatton methods first used to efficiently represent equivalence relations. The algorithms have O ((m+ n) a (m+ n, n)) running tunes, where m and n are measures of the problem size and a Is a functional reverse of Ackermann's function By usmg one or more of these algorithms m combination with other techniques, it is possible to solve the followmg graph problems m O (ma (m, n)) tnne, where m Is the number of edges and n Is the number of vertices m the problem graph