Generation of Trees, Two-Trees, and Storage of Master Forests

A new method of listing all possible trees of any given graph without duplication or redundancy, using simple geometrical properties of the graph, is proposed. The procedure given is suitable for both manual and automatic computation, and any modifications to the given graph can be catered to by suitable interpolation and extrapolation. An alternate method for complete graphs, derived from it, gives the trees arranged in an order most suitable for their storage as master forest matrices and for directly obtaining trees and 2-trees of any given graph through simple modifications to them instead of starting from scratch every time. Some properties of master forest matrices are discussed, which., inter alia, lead to a formula for the number of trees in a sub-graph of a complete graph. While an edge-numbering convention and a criterion for a tree play the key roles in systematic generation of trees, the storage technique makes it possible to obtain all the co-factors and determinants of a node-admittance matrix of any network (within the range of storage) by merely operating on one single master forest matrix.