#### Appendix B Segmenting woody stems

Following Fig. 2, the mass (*m*, kg) of a stem is:

- (Eqn B1)

(*m*_{q}, mass of liquid; *m*_{d}, mass of dry matter and the mass of internal air space is ignored.) The unitless ratio:

- (Eqn B2)

is usually called the moisture content in the wood science literature. Eqns B1 and B2 can be combined to give:

- (Eqn B3)

Our prime interest is in living trees so *α* would usually be greater than *α*_{f} (note that *α*_{f} is the fibre saturation point, see main text) and we restrict our derivations to that range. The mass concentration of dry matter ([*D*], kg m^{−3}) is:

- (Eqn B4)

(*V*, volume (m^{3}).) The density (*ρ*, kg m^{−3}) is:

- (Eqn B5)

The mass and volume of the structure (*m*_{s}, *V*_{s}) are:

- (Eqn B6a)

- (Eqn B6b)

(*ρ*_{c}, density of dry cell walls; *ρ*_{w}, density of liquid water and the swollen cell wall is assumed to be a linear mixture.) That latter assumption is only approximately true (Siau, 1984), but the results of the calculation are not overly sensitive to that assumption. We assume that *ρ*_{c} is 1.5 g cm^{−3}, and thereby ignore the effect of secondary compounds on the density of the dry cell walls (Siau, 1984; Skaar, 1988) which will lead to slight errors in some situations.

The mass and volume of the solution (*m*_{u}, *V*_{u}) are:

- (Eqn B6c)

- (Eqn B6d)

(*ρ*_{u}, density of the solution.) The volume of gas space (*V*_{a}) is given by:

- (Eqn B6e)

and the volumetric fraction of gas (*F*_{a}) is:

- (Eqn B6f)

The maximum value of *F*_{a} (*F*_{a:max}) occurs when *α* equals *α*_{f}, while the maximum moisture content (*α*_{max}) occurs when *F*_{a} is zero. Assuming that the density of the solution equals that of liquid water (*ρ*_{w}), it follows that:

- (Eqn B7a)

- (Eqn B7b)

- (Eqn B8a)

(units of [*D*], g cm^{−3}.) Note that *F*_{a:max} is also equal to the combined fractional volume of air space and solution, so Eqn B8a can also be written as:

- (Eqn B8b)

which is Eqn 6b in the main text. Observations show that *α*_{f} may be > 0.30 when [*D*] < 0.5 g cm^{−3} (Skaar, 1988). To take into account the dependence of *α*_{f} on [*D*], the following empirical relationship can be used (see Appendix C for derivation):

- (Eqn B9)

instead of adopting a constant value. The general nature of the relationship between *F*_{a:max} and [*D*] is unchanged by the use of Eqn B9 (Fig. B1), and in the analysis in the main text we used Eqn B8b.

In the normal course of events, *α* would fluctuate depending on leaf habit (e.g. deciduous trees) and weather conditions. The magnitude of those fluctuations can be estimated by the consequent variations in the density of the stem. Following the previous logic, the minimum density (*ρ*_{min}) would occur when *α* equals *α*_{f}, and the maximum density (*ρ*_{max}) would occur when *α* equals *α*_{max}. The resulting relationship between [*D*] and *ρ* (Fig. B2) shows that the day-to-day fluctuations should be larger in stems with a small [*D*].