String Coding of Trees with Locality and Heritability

We consider the problem of coding labelled trees by means of strings of vertex labels and we present a general scheme to define bijective codes based on the transformation of a tree into a functional digraph. Looking at the fields in which codes for labelled trees are utilized, we see that the properties of locality and heritability are required and that codes like the well known Prüfer code do not satisfy these properties. We present a general scheme for generating codes based on the construction of functional digraphs. We prove that using this scheme, locality and heritability are satisfied as a direct function of the similarity between the topology of the functional digraph and that of the original tree. Moreover, we also show that the efficiency of our method depends on the transformation of the tree into a functional digraph. Finally we show how it is possible to fit three known codes into our scheme, obtaining maximum efficiency and high locality and heritability.