A rectilinear Steiner tree for a set T of points in the plane is a tree which connects T using horizontal and vertical lines. In the RECTILINEAR STEINER TREE problem, input is a set T of n points in the Euclidean plane (ℝ2) and the goal is to find an rectilinear Steiner tree for T of smallest possible total length. A rectilinear Steiner arborecence for a set T of points and root r ∈ T is a rectilinear Steiner tree S for T such that the path in S from r to any point t ∈ T is a shortest path. In the RECTILINEAR STEINER ARBORESCENSE problem the input is a set T of n points in ℝ2, and a root r ∈ T, the task is to find an rectilinear Steiner arborescence for T, rooted at r of smallest possible total length. In this paper, we give the first subexponential time algorithms for both problems. Our algorithms are deterministic and run in 2O(√n log n) time. © Fedor Fomin, Sudeshna Kolay, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh.