  We introduce "$t$-LC triangulated manifolds" as those triangulations
obtainable from a tree of $d$-simplices by recursively identifying two boundary
$(d-1)$-faces whose intersection has dimension at least $d-t-1$. The $t$-LC
notion interpolates between the class of LC manifolds introduced by
Durhuus--Jonsson (corresponding to the case $t=1$), and the class of all
manifolds (case $t=d$). Benedetti--Ziegler proved that there are at most
$2^{d^2 \, N}$ triangulated $1$-LC $d$-manifolds with $N$ facets. Here we prove
that there are at most $2^{\frac{d^3}{2}N}$ triangulated $2$-LC $d$-manifolds
with $N$ facets. This extends to all dimensions an intuition by Mogami for
$d=3$.
  We also introduce "$t$-constructible complexes", interpolating between
constructible complexes (the case $t=1$) and all complexes (case $t=d$). We
show that all $t$-constructible pseudomanifolds are $t$-LC, and that all
$t$-constructible complexes have (homotopical) depth larger than $d-t$. This
extends the famous result by Hochster that constructible complexes are
(homotopy) Cohen--Macaulay.
