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

Contractible polyhedra in products of trees and absolute retracts in products of dendrites