We study a routing problem which occurs in high-speed (ATM) networks, termed the "rooted virtual path layout problem" on chain networks. This problem is essentially a tree embedding problem on a chain host graph. We present four performance measures for the quality of such an embedding which have practical implications, and find optimal solutions for each of them. We first show that the search can be restricted to the class of layouts with no crossovers. Given bounds on the load l and number of hops h in a layout, we then present a family of ordered trees T (ℓ, h), within which an optimal solution can be found (if one exists at all); this holds for either the worst-case or average-case measures, and for a chain of length n, with n ≤ (ℓ+hℓ). For the worst-case measures these trees are used in characterizing, constructing, and proving the optimality of the solutions. For each average-case measure, a recursive formulation of the optimal solution is presented, from which an optimal polynomial dynamic programming algorithm is derived. Furthermore, for the unweighted average measures, these formulations are explicitly solved, and the optimal solutions are mapped to T (ℓ, h).