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Reaction-diffusion-advection equation in binary tree networks and optimal size ratio

A simple reaction-diffusion-advection equation is proposed in a dichotomous tree network to discuss an optimal network. An optimal size ratio r is evaluated by the principle of maximization of total reaction rate. In the case of reaction-limited conditions, the optimal ratio can be larger than (1/2)^{1/3} for a fixed value of branching number, which is consistent with observations in mammalian lungs. We find furthermore that there is an optimal branching number when the Peclet number is large. Under the doubly optimal conditions with respect to the size ratio and branching number, the optimal value of r is close to (1/2)^{1/3}.