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 subgraphsof $G$ 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 beenundertaken on path decompositions by imposing certain conditionson the paths considered in the decomposition of the graph wherethe primary objective is to obtain the minimum number of pathsrequired for a certain type of decomposition for a given graph. Apath decomposition $psi$ such that the paths in $psi$ are inducedas well as of same parity is defined as an emph{equiparityinduced path decomposition}. The minimum number of paths in such adecomposition of a graph $G$ is called the emph{equiparity inducedpath 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.