Figure 2.

Algebraically equivalent representations of a Kirchhoff polynomial, their expression trees, and sizes for an example graph G (see electronic supplementary material, figure S1) in (a) the fully expanded representation, (b) a simplified representation, e.g. obtained by algorithm C_R and (c) a change of variables form (forest of expression trees), e.g. obtained by algorithm C_I. The size of a representation is the sum of the numbers of branch vertices and of leaves in the expression tree. With change of variables, each expression tree from the forest is assigned a pointer counting as 1 to the size of the representation and pointing to the leaves of other expression trees where it should be substituted to obtain the expression tree of the complete Kirchhoff polynomial. The pointer S denotes the ‘starting’ tree.