Equiparity induced path decomposition in trees

A emph{decomposition} quad of a graph quad $G$ is a collection $psi={H_{1}, H_{2}, ldots, H_{k}}$ of subgraphs of $G$ such that every edge of $G$ belongs to exactly one $H_{i}$. The decomposition $psi$ is called a emph{path decomposition} of $G$ if each $H_{i}$ is a path in $G$. Several studies have been undertaken on path decompositions by imposing certain conditions on the paths considered in the decomposition of the graph where the primary objective is to obtain the minimum number of paths required for a certain type of decomposition for a given graph. A path decomposition $psi$ such that the paths in $psi$ are induced as well as of same parity is defined as an emph{equiparity induced path decomposition}. The minimum number of paths in such a decomposition of a graph $G$ is called the emph{equiparity induced path decomposition number} of $G$ and is denoted by $pi_{pi}(G)$. In this paper we determine the value of $pi_{pi}$ for trees of even size.