Path-tables of trees: a survey and some new results

The (vertex) path-table of a tree $T$ contains quantitative information about the paths in $T$. The entry $(i,j)$ of this table gives the number of paths of length $j$ passing through vertex $v_i$. The path-table is a slight variation of the notion of path layer matrix. In this survey we review some work done on the vertex path-table of a tree and also introduce the edge path-table. We show that in general, any type of path-table of a tree $T$ does not determine $T$ uniquely. We shall show that in trees, the number of paths passing through edge $xy$ can only be expressed in terms of paths passing through vertices $x$ and $y$ up to a length of 4. In contrast to the vertex path-table, we show that the row of the edge path-table corresponding to the central edge of a tree $T$ of odd diameter, is unique in the table. Finally we show that special classes of trees such as caterpillars and restricted thin trees (RTT) are reconstructible from their path-tables