We show that a compact
n
n
-polyhedron PL embeds in a product of
n
n
trees if and only if it collapses onto an
(
n
−
1
)
(n-1)
-polyhedron. If the
n
n
-polyhedron is contractible and
n
≠
3
n\ne 3
(or
n
=
3
n=3
and the Andrews–Curtis Conjecture holds), the product of trees may be assumed to collapse onto the image of the embedding.
In contrast, there exists a
2
2
-dimensional compact absolute retract
X
X
such that
X
×
I
k
X\times I^k
does not embed in any product of
2
+
k
2+k
dendrites for each
k
k
.