Allometricscalingrelationsforhowbranchingtubesdeliverfluidareoftenbasedontheassumptionofisothermalconditionswhicharenotalwaysfulfilled.Thisworkreportsonthesizesofbranchingtubesforfluidflowsubjectedtoconstantwallheatflux.Itexploreshowthedeliveryoffluidandheatinfluencestheoptimalsizeofatubeandtheoptimalconfigurationofbranchingtubesforbothlaminarandturbulentflow.Besides,itpresentsananalyticalbasisforevaluationofoptimalsizeoftubesthatminimisethepowerlossassociatedwiththepumpingpowerandtheheattransfer.Thefindingsobtainedinthisstudymayhaveimportantimplicationsfordesignofsystemswithminimumthermodynamiclosses.