Let XP(/, //, []) be the fragment of XPath 1.0, consisting of queries that involve only the child and descendant axes, and predicates without disjunction or negation (and no wildcard nodetests); these queries can be represented as tree patterns. We consider the problem of rewriting a query Q using a materialized view V. where Q, V is an element of XP(/, //, []). We present more efficient algorithms for the following: (1) Determine if an equivalent rewriting of Q using V exists; find the smallest such rewriting, when it exists. A previously-known algorithm runs in O(vertical bar Q vertical bar(2) + vertical bar Q vertical bar vertical bar V vertical bar) time. For the special case when Q is known to be minimal, we present an O(vertical bar Q vertical bar vertical bar V vertical bar) algorithm. (2) Determine if a (nonempty) contained rewriting of Q using V exists. We present an O(vertical bar Q vertical bar vertical bar V vertical bar) algorithm, compared to the previous O(ver! tical bar Q vertical bar vertical bar V vertical bar(2)) algorithm. We also present a more efficient algorithm for finding a maximal such rewriting, when it exists. Then we extend this result to a subset of XP(/, //, [], *) that allows restricted occurrences of wildcard nodetests.