Obstacle-avoiding rectilinear Steiner tree (OARST) construction is a fundamental problem associated with the trend toward IP-block-based System-on-Chip designs. The objective of previous studies on obstacle-avoiding rectilinear Steiner minimal tree (OARSMT) has been to minimize the total wirelength of the constructed Steiner tree. Studies of performance-driven Steiner trees have demonstrated that the minimization of wirelength may worsen the performance of the Steiner tree. This work is the first to construct OARST while considering the Elmore delay. A critical-trunk-based tree growth mechanism is proposed. The critical trunks are constructed by extended single-source single-target maze routing called multi-source single-target maze routing. The unconnected pins are connected to critical trunks under the delay constraints of every sink. The proposed critical trunk can be applied to solve performance-driven and slack-driven OARST problems. Experimental results demonstrate that the proposed algorithms achieve an average 24.12% improvement in the maximum delay over OARSMT in performance-driven OARST problem and successfully solve 66.67% worst negative slack (WNS) violations in slack-driven OARST problem while running faster than previous OARSMT algorithms.