Davenport-Schinzel sequences are sequences that do not contain forbidden subsequences of alternating symbols. They arise in the computation of the envelope of a set of functions. It is shown that the maximal length of a Davenport-Schinzel sequence composed of n symbols is THETA (n alpha (n)), where alpha (n) is the functional inverse of Ackermann's function and is thus very slow growing. This is achieved by establishing an equivalence between such sequences and generalized path compression schemes on rooted trees and then analyzing these schemes.