Based on model-technical as well as physical considerations a nodal-point relation at bifurcations is proposed for one-dimensional (ID) network morphodynamic models: the ratio between the sediment transports into the downstream branches is proportional to a power of the discharge ratio. The influence of the nodal-point relation on the behaviour of the morphodynamic model is analyzed theoretically. The exponent in the nodal-point relation appears to be crucial for the stability of the bifurcation in the model. For large values of the exponent, the bifurcation is stable, i.e. the downstream branches remain open. For small values of the exponent, the bifurcation is unstable: only one of the branches tends to remain open. The exponent also has a strong influence on the morphological time scales of the network. The conclusions from the analysis have been verified by numerical simulations using a package for one-dimensional network modelling.