Losses due to irreversible processes accompanying the heat and fluid flows. This paper addresses the fundamental question of what the shape of parent and offspring branches should be, to minimize the total irreversibility. I explore how the delivery of fluid and heat influences both the optimal size of Y-assemblies of tubes and Y-shaped junctions of tubes filled with a packing material, that are used as an elemental unit to construct tree-shaped flow networks. The geometric features of these constructs are obtained from the global minimization of expenditure of power associated to fluid flow and heat transfer. The optimal step factors for tube diameters and for tube lengths, and the optimal angle between the branches are obtained under different size-limiting constraints (surface area and volume). Among other results, I show that the geometrical features of the construct may be influenced by the heat flux. Besides, it is crucial to include the size-limiting requirements in agreement with goal of the system. For similar geometries, the heat and mass transfer correlations are analogous. Then, optimal size ratios and optimal angles of branching obtained here are also valid for mass flux by replacing the heat transfer dimensionless numbers by the equivalent mass transfer numbers. The analytical expressions obtained here provide fundamental information for designers assemble the inserts that formed a tree-shaped flow network. (C) 2018 Elsevier Ltd. All rights reserved.