By Computational Algebraic Topology we mean an area of Computational Mathematics dealing with the effective and efficient construction of algebraic topological invariants for discrete or continuous ob jects. Eilenberg and Mac Lane in the six-ties of the last century develop a framework for Algebraic Topology mainly based on the notions of simplicial sets and chain homotopy equivalences. This setting which gives preference to chain homotopy operators in combinatorial ambiance has been followed in computational theories like Effective Homology, Homological Perturbation Theory and Discrete Morse Theory.  Here, we analyze the high data redundancy existing in chain homotopyoperators and we design a method for describing these maps in terms of  of trees (homological trees) in which we remove this excess of non-essential algebraic information.