  In this paper we consider alignment of sparse graphs, for which we introduce
the Neighborhood Tree Matching Algorithm (NTMA). For correlated
Erd\H{o}s-R\'{e}nyi random graphs, we prove that the algorithm returns -- in
polynomial time -- a positive fraction of correctly matched vertices, and a
vanishing fraction of mismatches. This result holds with average degree of the
graphs in $O(1)$ and correlation parameter $s$ that can be bounded away from 1,
conditions under which random graph alignment is particularly challenging. As a
byproduct of the analysis we introduce a matching metric between trees and
characterize it for several models of correlated random trees. These results
may be of independent interest, yielding for instance efficient tests for
determining whether two random trees are correlated or independent.
