## Importance of the conduit furcation number f

A final distinction between animal and plant vasculature concerns the daughter to mother ratio in conduit number. In animals where the network consists of a single open tube that branches, F a 2. In plants, the network at every level from trunk to twig consists of numerous conduits running in parallel and F > 0. The increase in the number of tree branches moving from trunk to twig occurs independently of the change in the number of conduits in each branch. For any possible F, however, the optimal (maximum) conductivity occurs when the 2r3 is conserved (Figure 4.2A, Ref. [11]).

The value of the conduit furcation number has adaptive significance for plants. The "Murray's law optimum" line in Figure 4.2B shows the conductance of a Murray's law network (2r3 conserved across three bifurcating branch ranks) that corresponds to different values of F when all else (volume and branching pattern) is held constant. The absolute value of the hydraulic conductance increases with F [11]. The increase in efficiency with F is greater with an increasing number of branch ranks in the network; Figure 4.2 is for a relatively small network of three branch ranks. Conductance increases because larger F means the trunk and major branches have fewer but wider and more conductive conduits for a given number of conduits in the twigs (held to the minimum of one per twig for the reference network in Figure 4.2). The plant becomes more like a branching aorta, which is the peak of efficiency. Thus, for the same investment in conduit volume, plants can increase their hydraulic conductance by having more conduits running in parallel in their twigs than in their trunk while also conforming to Murray's law.

The West et al. model assumed F = 1, citing the pipe model of tree form [32,33]. However, the "unit pipe" associated with each leaf in the pipe model does not refer to an actual fluid conducting pipe but rather to a strip of wood of constant cross-sectional area. The pipe model does not specify the anatomical composition of this strand of wood — how much of it is fibers, parenchyma, conduits, and so forth — because it is only concerned with biomass allocation and not hydraulics. There is no a priori value of F in plants except that it must be greater than zero.

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