  Individualization-Refinement (IR) algorithms form the standard method and
currently the only practical method for symmetry computations of graphs and
combinatorial objects in general. Through backtracking, on each graph an
IR-algorithm implicitly creates an IR-tree whose order is the determining
factor of the running time of the algorithm.
  We give a precise and constructive characterization which trees are IR-trees.
This characterization is applicable both when the tree is regarded as an
uncolored object but also when regarded as a colored object where vertex colors
stem from a node invariant. We also provide a construction that given a tree
produces a corresponding graph whenever possible. This provides a constructive
proof that our necessary conditions are also sufficient for the
characterization.
