  The correlated Erd\"os-R\'enyi random graph ensemble is a probability law on
pairs of graphs with $n$ vertices, parametrized by their average degree
$\lambda$ and their correlation coefficient $s$. It can be used as a benchmark
for the graph alignment problem, in which the labels of the vertices of one of
the graphs are reshuffled by an unknown permutation; the goal is to infer this
permutation and thus properly match the pairs of vertices in both graphs. A
series of recent works has unveiled the role of Otter's constant $\alpha$ (that
controls the exponential rate of growth of the number of unlabeled rooted trees
as a function of their sizes) in this problem: for $s>\sqrt{\alpha}$ and
$\lambda$ large enough it is possible to recover in a time polynomial in $n$ a
positive fraction of the hidden permutation. The exponent of this polynomial
growth is however quite large and depends on the other parameters, which limits
the range of applications of the algorithm. In this work we present a family of
faster algorithms for this task, show through numerical simulations that their
accuracy is only slightly reduced with respect to the original one, and
conjecture that they undergo, in the large $\lambda$ limit, phase transitions
at modified Otter's thresholds $\sqrt{\widehat{\alpha}}>\sqrt{\alpha}$, with
$\widehat{\alpha}$ related to the enumeration of a restricted family of trees.
